b10a

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Statistics

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Feb 20, 2024

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16.6 a. False. The distribution of sample proportions is not necessarily right skewed for random samples of size 12. b. True. To have an approximately normal distribution of sample proportions, the sample size generally needs to be at least 40. c. False. A random sample of 50 young Americans with a 20% delay in starting a family is not automatically considered unusual. d. True. A random sample of 150 young Americans with a 20% delay in starting a family would be considered unusual. e. False. Tripling the sample size does not reduce the standard error of the sample proportion by exactly one-third, but rather by roughly one-third. 16.20 a. The population parameter of interest is the proportion of US adults who favor requiring proof of COVID-19 vaccination for travel by airplane, and the point estimate is 0.57. b. To check the conditions required for constructing a confidence interval using a mathematical model, you have to ensure that the sample is random, the sample size is sufficiently large, and the sampling distribution is approximately normal. Since the Gallup poll states that the sample was randomly sampled, and the sample size is 3,731 we can assume that the conditions are met. c. The 95% confidence interval for the proportion of US adults who favor requiring proof of COVID-19 vaccination for travel by airplane can be calculated using the point estimate and the margin of error. d. A higher confidence level would result in a wider confidence interval. e. A larger sample size would result in a narrower confidence interval.

16.22 a. 60% is a sample statistic because it is based on the responses of the random sample of 1,563 US adults. b. The 95% confidence interval for the proportion of US adults who think marijuana should be made legal is 0.60 ± (1.96 * √((0.60 * 0.40) / 1563)). c. We cannot definitively determine if the normal model is a good approximation for these data. d. Based on the confidence interval, we can say with 95% confidence that a majority of US adults think marijuana should be legalized if the interval does not include 50%. 16.30 To construct a 95% confidence interval for the fraction of all shoppers during the year whose visit was due to a coupon received in the mail, we use this formula CI = 0.2356 ÷ (1.96 * /((0.2356 * (1 - 0.2356)) / 603)) Which simplifies down to CI = 0.2356 ‡ 0.0339 So the 95% confidence interval for the fraction of all shoppers during the year whose visit was due to a coupon received in the mail is approximately 0.2017 to 0.2695. 19.4 a. n = 10: 2.97 centimeters b. n = 50: 1.33 centimeters

c. n = 100: 0.94 centimeters d. n = 1000: 0.30 centimeters e. The standard error of the mean is a number that describes the variability or uncertainty in the sample mean. It indicates how much the sample mean is likely to vary from the population mean. A smaller standard error of the mean indicates a more precise estimate of the population mean. 19.8 The dashed curve represents the t-distribution with 1 degree of freedom. The dotted curve represents the t-distribution with 5 degrees of freedom. The solid curve represents the standard normal (z) distribution. 19.13 a. False. The correct interpretation is that we are 95% confident that the true population average falls between 440 and 520 cans of soda per year. This does not mean that 95% of adults in the US consume between 440 and 520 cans of soda per year. The confidence interval only provides a range of values within which the population average is likely to fall. b. False. The correct interpretation is that if we were to repeat the sampling and construct 95% confidence intervals each time, 95% of those intervals would contain the true population average. Once the interval is constructed, it either contains the true population average or it does not. The probability applies to the construction of intervals, not to the true population parameter itself. c. True. This statement accurately represents the interpretation of the confidence interval. We are 95% confident that the true population average per adult yearly soda consumption is between 440 and 520 cans. d. False. The correct interpretation is that we are 95% confident that the true population average falls between 440 and 520 cans of soda per year, not that the average soda consumption of the sampled individuals falls within this range. The confidence interval provides information about the population parameter, not the specific sampled individuals. 19.16

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Sample mean = (65 + 77) / 2 = 71 Margin of error = (77 - 65) / 2 = 6 Sample standard deviation = 6 / (1.711 * /25) = 6 / (1.711 * 5) = 6 / 8.555 = 0.701 So as we see the sample mean is 71, the margin of error is 6, and the sample standard deviation is 0.701.

b10a

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