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Find the probability that more than 155 favor a charter school. d. Find the probability that fewer than 147 favor a charter school. e. Find the probability that exactly 175 favor a charter school. Let X = the number that favor a charter school for grades K through 5. X ~ B(n, p) where n = 300 and p = 0.53. Because np ... |
distribution. If you type in binomial probability distribution calculation in an internet browser, you can find at least one online calculator for the binomial. For Example 7.10, the probabilities are calculated using the following binomial distribution: (n = 300 and p = 0.53). Compare the binomial and normal distribu... |
): a. b. x¯ = _______ s = _______ 5. Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 7 | The Central Limit Theorem 433 Collecting Avera... |
__________ Table 7.3 3. Construct a histogram. Scale the axes using the same scaling you used for the section titled Collect the Data. Sketch the graph using a ruler and a pencil. Figure 7.12 4. Calculate the following (n = 5, surveying five people at a time): a. b. x¯ = _______ s = _______ 5. Draw a smooth curve throu... |
orem Sample 1 Sample 2 Sample 3 Sample 4 Sample means from other groups: Means: x¯ = ____ x¯ = ____ x¯ = ____ x¯ = ____ Table 7.5 2. Calculate the following: a. b. x¯ = _______ s x¯ = _______ 3. Again, use a random number generator to randomly select four samples from the population. This time, make the samples of size... |
Discussion Questions 1. Compare the three histograms you have made, the one for the population and the two for the sample means. In three to five sentences, describe the similarities and differences. 2. State the theoretical (according to the clt) distributions for the sample means. a. n = 5: x¯ ~ _____(_____, _____) ... |
= 1 m the cumulative distribution function is P(X ≤ x) = 1 – e–mx mean a number that measures the central tendency; a common name for mean is average; the term mean is a shortened form of arithmetic mean;. by definition, the mean for a sample (denoted by x¯ ) is x¯ = sum of all values in the sample number of values in... |
limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution, even if the... |
Assume that the 16 reviews represent a random set of reviews. 1. What is the mean, standard deviation, and sample size? 2. Complete the distributions. a. X ~ _____(_____, _____) ¯ ~ _____(_____, _____) b. X This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 7 | The Central Limit Th... |
the sum that is 1.5 standard deviations below the mean of the sums. 15. Find the percentage of sums between 1.5 standard deviations below the mean of the sums and one standard deviation above the mean of the sums. Use the following information to answer the next six exercises: A researcher measures the amount of sugar... |
with a standard deviation of 0.7. She samples 400 customers. 31. What is the z-score for Σx = 840? 32. What is the z-score for Σx = 1,186? 33. What is P(Σx < 1186)? Use the following information to answer the next three exercises: An unkwon distribution has a mean of 100, a standard deviation of 100, and a sample size... |
batteries last? 50. What is the distribution for the total length of time 64 batteries last? 51. Find the probability that the sample mean is between 7 and 11. 52. Find the 80th percentile for the total length of time 64 batteries last. 53. Find the interquartile range (IQR) for the mean amount of time 64 batteries la... |
average of 49 fly balls. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 7 | The Central Limit Theorem 445 63. According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS For... |
. In words, X ¯ ~ _____(_____, _____) ¯ d. The IQR for X c. X is from _______ acres to _______ acres. 67. Determine which of the following are true and which are false. Then, in complete sentences, justify your answers. ¯ a. When the sample size is large, the mean of X is approximately equal to the mean of Χ. ¯ b. When... |
0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. The distribution to use for the average cost of gasoline for the 16 gas stations is: a. X b. X c. X d. X ¯ ~ N(4.59, 0.10) ¯ ~ N ⎛ ⎝4.59, 0.10 16 ⎝4.59, 16 0.10 ⎝4.59, 16 0.10 ¯ ~... |
X = _____________. a. b. X ~ _____(_____, _____) c. d. ΣX ~ _____(_____, _____) e. Find the probability that the managers earn a total of over $400,000. f. Find the 90th percentile for an individual manager's salary. g. Find the 90th percentile for the sum of ten managers' salary. h. i. If we surveyed 70 managers inste... |
7 29.25 a. b. In words, Χ = ______________. x¯ = _____ i. ii. sx = _____ iii. n = _____ c. Construct a histogram of the distribution of the averages. Start at x = –0.0005. Use bar widths of 10. d. e. Randomly average five stock prices together. (Use a random number generator.) Continue averaging five prices In words, d... |
be surprised, based on numerical calculations, if the sample average wait time (in minutes) for 100 riders was less than 30 minutes? a. yes b. no c. There is not enough information. Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution w... |
60, 9). Suppose that you form random samples of 25 from this distribution. Let X averages. Let ΣX be the random variable of sums. For parts (c) through (f), sketch the graph, shade the region, label and scale the horizontal axis for X ¯, and find the probability. ¯ on the same graph. a. Sketch the distributions of X an... |
for the average manager's salary. 90. The average length of a maternity stay in a U.S. hospital is said to be 2.4 days with a standard deviation of 0.9 days. We randomly survey 80 women who recently bore children in a U.S. hospital. a. b. In words, X = _____________. ¯ = ___________________. In words, X ¯ ~ _____(____... |
(g) Yellow (g) Brown (g) Blue (g) Green (g) 0.883 0.769 0.859 0.784 0.824 0.858 0.848 0.851 0.696 0.876 0.855 0.806 0.840 0.868 0.859 0.982 0.751 0.841 0.856 0.799 0.966 0.859 0.857 0.942 0.873 0.809 0.890 0.878 0.905 0.735 0.895 0.865 0.864 0.852 0.866 0.859 0.838 0.863 0.888 0.925 0.793 0.977 0.850 0.830 0.856 0.842... |
7 | The Central Limit Theorem 453 94. The Screw Right Company claims their 3 4 inch screws are within ±0.23 of the claimed mean diameter of 0.750 inches with a standard deviation of 0.115 inches. The following data were recorded. 0.757 0.723 0.754 0.737 0.757 0.741 0.722 0.741 0.743 0.742 0.740 0.758 0.724 0.739 0.736... |
an, D. (2010). 20 percent of Americans have never used email. WebGuild. Retrieved from http://www.webguild.org/ 20080519/20-percent-of-americans-have-never-used-email The Flurry Blog. (2013). Retrieved from http://blog.flurry.com U.S. Department of Agriculture. (n.d.). Retrieved from https://www.usda.gov/ 7.2 The Centr... |
carry This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 7 | The Central Limit Theorem 455 b. Χ ~ E(0.88, 0.88) ¯ = average amount of change carried by a sample of 25 students. c. X ¯ ~ N(0.88, 0.176) d. X e. 0.0819 f. 0.1882 g. The distributions are different. Part (a) is exponent... |
(189, 21) c. 0.0432 456 Chapter 7 | The Central Limit Theorem d. 162.09; 90 percent of the total nine trials of this type will last 162 days or more. 75 a. X = the salary of one elementary school teacher in the district b. X ~ N(44000, 6500) c. ΣX ~ sum of the salaries of 10 elementary school teachers in the sample d. ... |
calculate the probability, we use normalcdf(lower, upper, μ, σ n ) = normalcdf ⎛ ⎝E – 99,16.7,17, 0.8 30 ⎞ ⎠ = 0.0200. • If the process is working properly, then the probability that a sample of 30 batteries would have at most 16.7 life span hours is only 2%. Therefore, the class was justified to question the claim. 9... |
confidence intervals for estimating a population mean and a population proportion • • Discriminate between problems applying the normal and the Student's t-distributions • Calculate the sample size required to estimate a population mean and a population proportion, given a desired confidence level and margin of error ... |
, but we do know that the population standard deviation is σ = 1 and our sample size is 100. Then, by the central limit theorem, the standard deviation for the sample mean is σ n = 1 100 = 0.1. The empirical rule, which applies to bell-shaped distributions, says that in approximately 95 percent of the samples, the samp... |
sample produced an x¯ all the samples (95–100 percent). that is not within 0.2 units of the true mean μ. The second possibility happens for only 5 percent of Remember that a confidence interval is created for an unknown population parameter like the population mean, μ. This OpenStax book is available for free at http:... |
mean, x¯, is the point estimate of the unknown population mean, μ. The confidence interval (CI) estimate will have the form: (point estimate – error bound, point estimate + error bound) or, in symbols, ( x¯ – EBM, x¯ +EBM ). The margin of error (EBM) depends on the confidence level (CL). The confidence level is often ... |
.645 is the z-score in a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. To capture the central 90 percent, we must go out 1.645 standard deviations on either side of the calculated sample mean. The critic... |
BM and construct the confidence interval. We need to find the value of z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z ~ N(0, 1). The confidence level, CL, is the area in the middle of the standard normal distribution. CL = 1 – α, so α is the area ... |
should clearly state the confidence level (CL), explain which population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints): "We estimate with ___percent confidence that the true population mean (include the context of the problem) is between ___ and ___ (include... |
8225). Solution 8.2 Solution B Press STAT and arrow over to TESTS. Arrow down to 7:ZInterval. Press ENTER. Arrow to Stats and press ENTER. Arrow down and enter 3 for σ, 68 for x¯, 36 for n, and.90 for C-level. Arrow down to Calculate and press ENTER. The confidence interval is (to three decimal places)(67.178, 68.822).... |
for the true (population) mean of the SARs for cell phones. Assume that the population standard deviation is σ = 0.337. Solution 8.3 Solution A To find the confidence interval, start by finding the point estimate: the sample mean, x¯ = 1.024. This is calculated by adding the specific absorption rate for the 30 cell ph... |
1150 1250 1350 Table 8.2 1.48 1450 0.8 1.15 1.36 0.77 1550 1650 1750 1850 0.462 1950 1.36 1.39 1.3 0.7 2050 2150 2250 2350 1.53 0.68 1.4 1.24 0.57 0.2 0.51 0.3 0.73 0.869 Notice the difference in the confidence intervals calculated in Example 8.3 and the following Try It exercise. These intervals are different for sev... |
+ 0.98 = 68.98 Notice that the EBM is larger for a 95 percent confidence level in the original problem. Interpretation We estimate with 95 percent confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98. Explanation of 95 percent Confidence Level 95 percent of all confidence ... |
25, we increase the error bound. When n = 25, EBM = ⎛ ⎝1.645) ⎛ ⎝ ⎞ ⎠ 3 25 = 0.987. Summary: Effect of Changing the Sample Size • Increasing the sample size causes the error bound to decrease, making the confidence interval narrower. • Decreasing the sample size causes the error bound to increase, making the confidenc... |
margin of error is half of the length of the interval. Calculate the sample mean: • • If we know the error bound, x¯ = 68.82 – 0.82 = 68. If we do not know the error bound, x¯ = (67.18 + 68.82) 2 = 68. 8.6 Suppose we know that a confidence interval is (42.12, 47.88). Find the error bound and the sample mean. Calculati... |
, we rarely know the population standard deviation. In the past, when the sample size was large, this unknown number did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close-enough results. However, s... |
), compare to the standard normal distribution (in black). This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 8 | Confidence Intervals 473 Figure 8.6 • The exact shape of the Student's t-distribution depends on the degrees of freedom. As the degrees of freedom increase, the graph of... |
probability. The table is very limited. Calculators and computers can easily calculate any Student's t-probabilities. If the population standard deviation is not known, the error bound for a population mean is • EBM = ⎛ ⎝ is the t-score with area to the right equal to α 2, • use df = n – 1 degrees of freedom, and • s ... |
0.025 The area to the right of t0.025 is 0.025, and the area to the left of t0.025 is 1 – 0.025 = 0.975. = t0.025 = 2.14 using invT(.975,14) on the TI-84+ calculator. t α 2 EBM = ⎛ ⎝ ⎛ EBM = (2.14) ⎝ x¯ – EBM = 8.2267 – 0.9240 = 7.3 ⎞ ⎠ = 0.924 1.6722 15 This OpenStax book is available for free at http://cnx.org/conten... |
group of researchers is working to understand the scope of industrial pollution in the human body. Industrial chemicals may enter the body through pollution or as ingredients in consumer products. In October 2008, the scientists tested cord-blood samples for 20 newborn infants in the United States. The cord blood of t... |
and 137.488. Solution 8.9 Solution B Enter the data as a list. Press STAT and arrow over to TESTS. Arrow down to 8:TInterval and press ENTER (or you can just press 8). Arrow to Data and press ENTER. Arrow down to List and enter the list name where you put the data. Arrow down to Freq and enter 1. Arrow down to C-level... |
or No. Examples of situations where you are the following trying to estimate the true population proportion are the following: What proportion of the population smoke? What proportion of the population will vote for candidate A? What proportion of the population has a college-level education? The distribution of the s... |
randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes, they own cell phones. Using a 95 percent confidence level, compute a confidence interval estimate for the true proportion of adult residents of this city who have cell ... |
phones. Explanation of 95 percent Confidence Level Ninety-five percent of the confidence intervals constructed in this way would contain the true value for the population proportion of all adult residents of this city who have cell phones. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8... |
�� p′ q′ n = (1.645) (0.60)(0.40) 500 = 0.036 p′ – EBP = 0.60 − 0.036 = 0.564 p′ + EBP = 0.60 + 0.036 = 0.636 480 Chapter 8 | Confidence Intervals The confidence interval for the true binomial population proportion is (p′ – EBP, p′ + EBP) = (0.564, 0.636). Interpretation • We estimate with 90 percent confidence that th... |
then, is n + 4, and the new count of successes is x + 2. Computer studies have demonstrated the effectiveness of the plus-four confidence interval for p method. It should be used when the confidence level desired is at least 90 percent and the sample size is at least ten. This OpenStax book is available for free at ht... |
reflect these additional trials. 8.12 Out of a random sample of 65 freshmen at State University, 31 students have declared their majors. Use the plus-four method to find a 96 percent confidence interval for the true proportion of freshmen at State University who have declared their majors. 482 Chapter 8 | Confidence I... |
having more than 500 online friends based on this larger sample. Compare the results to those in Example 8.13. Calculating the Sample Size n If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. This OpenStax book is available for free at htt... |
percentage of customers who click on ads on their smartphones. How many customers should the company survey in order to be 90 percent confident that the estimated proportion is within 5 percentage points of the true population proportion of customers who click on ads on their smartphones? Assume that the sample propor... |
. Figure 8.7 5. Some students think that a 90 percent confidence interval contains 90 percent of the data. Use the list of data on the first page and count how many of the data values lie within the confidence interval. What percentage is this? Is this percentage close to 90 percent? Explain why this percentage should ... |
has not taken statistics. 2. In one to two complete sentences, explain what this confidence interval means for this particular study. 3. Construct a confidence interval for each confidence level given. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 8 | Confidence Intervals 487 ... |
Women (in Inches) 1. Table 8.8 lists the heights of 100 women. Use a random number generator to select 10 data values randomly. 2. Calculate the sample mean and the sample standard deviation. Assume that the population standard deviation is known to be 3.3 in. With these values, construct a 90 percent confidence inter... |
of women) and count how many of the data values lie within the confidence interval that you generated based on that data. How many of the 100 data values lie within your confidence interval? What percentage is this? Is this percentage close to 90 percent? 7. Explain why it does not make sense to count data values that... |
a sample statistic For example, if four out of the 100 calculators sampled are defective, we might infer that 4 percent of the production is defective. normal distribution a bell-shaped continuous random variable X, with center at the mean value (μ) and distance from the center to the inflection points of the bell cur... |
as well. As the sample size increases, the EBM decreases. By the central limit theorem, EBM = z σ n. Given a confidence interval, you can work backward to find the error bound (EBM) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the ... |
Let p′ represent the sample proportion, x/n, where x represents the number of successes, and n represents the sample size. Let q′ = 1 – p′. Then the confidence interval for a population proportion is given by the following formula: (lower bound, upper bound) = (p′ – EBP, p′ + EBP) = (p′ – z p′ q′ n, p′ + z p′ q′ n ). ... |
score, which measures how far away a measure is from the population mean in the Student’s t-distribution. df = n – 1; t-distribution, where n represents the size of the sample the degrees of freedom for a Student’s T~tdf the random variable, T, has a Student’s t-distribution with df degrees of freedom EBM = t α 2 s n =... |
interval for a proportion: (lower bound, upper bound) = (p′ – EBP, p′ + EBP) = ⎛ ⎝p′ – z p′ q′ n, p′ + z p′ q′ n ⎞ ⎠ n = 2 p′ q′ z α 2 EBP2 provides the number of participants needed to estimate the population proportion with confidence 1 – α and margin of error EBP. PRACTICE Use the normal distribution for a single p... |
State the confidence interval, sketch the graph, and calculate the error bound. 10. If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make? 11. If the Census did another survey, kept the error bound the same, and surveyed only 50 ... |
¯ = _____ 24. n = _____ 25. ________ = 15 ¯. 26. In words, define the random variable X 27. What is x¯ estimating? 28. Is σ x known? 29. As a result of your answer to Exercise 8.26, state the exact distribution to use when calculating the confidence interval. Construct a 95 percent confidence interval for the true mean... |
were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watch an average of 151 hours each month, with a standard deviation of 32 hours. Assume that the underlying population distribution is normal. 43. Identify the following: a. b. x¯ =_______ s x =_______ c... |
instead of 39. Would the error bound become larger or smaller? How do you know? 61. Using the same x¯, s x, and n = 39, how would the error bound change if the confidence level were reduced to 90 percent? Why? 8.3 A Population Proportion Use the following information to answer the next two exercises: Marketing compani... |
we want to lower the sampling error. What is one way to accomplish that? 73. The sampling error given in the survey is ±2 percent. Explain what the ±2 percent means. Use the following information to answer the next five exercises: A poll of 1,200 voters asked what the most significant issue was in the upcoming electio... |
, upper and lower limits of the confidence interval, and the sample proportion. Figure 8.11 91. In one complete sentence, explain what the interval means. 92. Using the same p′ and level of confidence, suppose that n were increased to 100. Would the error bound become larger or smaller? How do you know? 93. Using the s... |
iii. n =________ ¯. In words, define the random variables X and X b. c. Which distribution should you use for this problem? Explain your choice. d. Construct a 90 percent confidence interval for the population mean time to complete the tax forms. i. State the confidence interval. ii. Sketch the graph. iii. Calculate t... |
is 7.9, with a sample standard deviation of 2.8. x¯ =________ a. i. ii. σ =________ iii. n =________ ¯ b. Define the random variables X and X c. Which distribution should you use for this problem? Explain your choice. d. Construct a 90 percent confidence interval for the population mean number of letters campers send ... |
$6,600 $202,400 $15,800 Table 8.11 a. Find the point estimate for the population mean. b. Using 95 percent confidence, calculate the error bound. c. Create a 95 percent confidence interval for the mean total individual contributions. d. Interpret the confidence interval in the context of the problem. 102. The American... |
414, 7,681, 3,200, 17,500, 9,200, 7,380, 18,314, 6,557, 13,713, 17,768, 7,493, 2,771, 2,861, 1,263, 7,285, 28,165, 5,080, 11,622. Assume the underlying population is normal. a. i. ii. x¯ = __________ s x = __________ iii. n = __________ iv. n – 1 = __________ ¯ b. Define the random variables X and X c. Which distributi... |
__________ iv. n – 1 = __________ b. Define the random variable X in words. ¯ c. Define the random variable X d. Which distribution should you use for this problem? Explain your choice. e. Construct a 95 percent confidence interval for the population mean length of time. in words. i. State the confidence interval. ii. ... |
$76,530.80 $119,459.20 $0 $63,520.00 $6,500.00 $502,578.00 $705,061.10 $708,258.90 $135,810.00 $2,000.00 $2,000.00 $0 $1,287,933.80 $219,148.30 Table 8.12 x¯ = $251, 854.23 s = $521, 130.41 Use the sample data to construct a 96 percent confidence interval for the mean amount of money raised by all Leadership PACs duri... |
sample of 84 used car sales costs, the sample mean was $6,425, with a standard deviation of $3,156. Assume the underlying distribution is approximately normal. a. Which distribution should you use for this problem? Explain your choice. ¯ b. Define the random variable X c. Construct a 95 percent confidence interval for... |
which percentage of the confidence intervals constructed should contain the population mean worth of coupons? Explain why. Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16-oz... |
people who believe the president is doing an acceptable job. a. Define the random variables X and P′ in words. b. Which distribution should you use for this problem? Explain your choice. c. Construct a 90 percent confidence interval for the population proportion of people who believe the president is doing an acceptab... |
the random variables X and P′ in words. b. Which distribution should you use for this problem? Explain your choice. c. Construct a 97 percent confidence interval for the population proportion of people over 50 who ran and died in the same 8-year period. i. State the confidence interval. ii. Sketch the graph. iii. Calc... |
that education and our schools is one of the top issues facing California. We wish to construct a 90 percent confidence interval for the true proportion of California adults who believe that education and the schools is one of the top issues facing California. 125. A point estimate for the true population proportion i... |
any differences Identify CL and α. between the values. f. Create a confidence interval for the results of this study. g. A reporter is covering the release of this study for a local news station. How should she explain the confidence interval to her audience? 131. A national survey of 1,000 adults was conducted on May... |
interval for the proportion of town residents who think winterproofing their cars is very important. Explain what this confidence interval means in the context of this scenario. REFERENCES 510 Chapter 8 | Confidence Intervals 8.1 A Single Population Mean Using the Normal Distribution Centers for Disease Control and Pr... |
d.). Retrieved from http://www.businessweek.com/ Environmental Working Group. (n.d.). Human toxome project: Mapping the pollution in people. Retrieved from http://www.ewg.org/sites/humantoxome/participants/participant-group.php?group=in+utero%2Fnewborn Federal Election Commission. (n.d.). Disclosure data catalog: 2012 ... |
This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 8 | Confidence Intervals 511 Zogby Analytics. (2013). New SUNYIT/Zogby analytics poll: Few Americans worry about emergency situations occurring in their community; Only one in three have an emergency plan; 70% support infrastructur... |
a smaller interval) to capture the true population mean. ¯ 39 X is the number of hours a patient waits in the emergency room before being called back to be examined. X mean wait time of 70 patients in the emergency room. is the 41 CI: (1.3808, 1.6192) Figure 8.14 EBM = 0.12 43 a. b. x¯ = 151 s x = 32 This OpenStax boo... |
night beginning ice-skating class 81 a. x = 64 b. n = 80 c. p′ = 0.8 83 p 85 P′ ~ N ⎛ ⎝0.8, (0.8)(0.2) 80 ⎞ ⎠ CI = (0.72171, 0.87829). 87 0.04 89 (0.72; 0.88) 91 With 92 percent confidence, we estimate the proportion of girls, ages 8 to 12, in a beginning ice-skating class at the Ice Chalet to be between 72 percent an... |
. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval. g. According to the error bound formula, the firm needs to survey 206 people. Because we increase the confidence level, we need to increase either our error bound or the sample size. 99 a. i. 7.9 i... |
the problem, you know that σ = 2.5, and you need EBM = 1. z = z0.035 = ≈ 20.52. You need to measure at least 21 male students to achieve your goal. the density curve to the right of z is 0.035.) 1.812. n = z2 σ 2 EBM 2 = 1.8122 2.52 12 105 a. i. 8,629 ii. 6,944 iii. 35 iv. 34 t34 i. CI: (6244, 11,014) b. c. 518 Chapte... |
percent confidence that the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle lies between $47,292.57 and $456,415.89. Alternate Solution STATTESTS8:TIntervalENTERENTERFreqC-LevelCalculateEnter The difference between solutions arises from rounding differences. 111a. This OpenStax b... |
a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families. d. We can say that there is a significant difference between the... |
with x = (0.52)(1,000), n = 1,000, CL = 0.75. Answer is (0.502, 0.538). This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 8 | Confidence Intervals 521 d. Yes, this interval does not fall below 0.50, so we can conclude that at least half of all American adults believe that major sp... |
women managers in their company earn an average of $60,000 per year. A statistician will make a decision about these claims. This process is called hypothesis testing. A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is... |
than (>) Table 9.1 NOTE H0 always has a symbol with an equal in it. Ha never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with This OpenStax book is available for free at http://cnx.org/c... |
4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H0: p ≤ 0.066 Ha: p > 0.066 9.4 On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the fi... |
Hypothesis Testing with One Sample 527 The Power of the Test is 1 – β. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test. The following are examples of Type I and Type II errors. Example 9.5 Suppose the null hypothesis, H0, is: Frank's rock... |
II? Example 9.7 It’s a Boy Genetic Labs, a genetics company, claims to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, H0, is: It’s a Boy 528 Chapter 9 | Hypothesis Testing with One Sample Genetic Labs has no e... |
is less than 75 percent. In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75 percent of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option. 9.8 Determine both Type I and... |
The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known. When you perform a hypothesis test of a single population proportion p, you take a simple random sample from the population. You ... |
wrong and there are more $100 bills in the basket. A rare event has occurred (Didi getting the $100 bill) so Ali doubts the assumption about only one $100 bill being in the basket. Using the Sample to Test the Null Hypothesis After you collect data and obtain the test statistic (the sample mean, sample proportion, or ... |
value, then, is the probability that a sample mean is the same or greater than 17 cm when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for means. In Figure 9.2, the p-value is the area under the normal curve to the right of 17. Using a normal distribution table... |
data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of H0. Conclusion: After you make your decision, write a thoughtful conclusion about the hypotheses in terms of the given problem. Example 9.10 When using the p-value to evaluate a hypothesis test, you might find it useful to ... |
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