Answer Key
University:
Stanford UniversityCourse:
MATH 20 | CalculusAcademic year:
2023
Views:
24
Pages:
16
Author:
MathPaladin
0.6699 Which indicates that we are 99% confident that the true population proportion is contained by the intervals (0.4901, 0.6699). Question #9 The number of points TEDU b etball team scores against Stanford b etball team is normally distributed. Suppose that over the last 10 games between the two teams TEDU scored the points given below: a) Construct a 95% two-sided confidence interval for the mean of the number of points scored by TEDU. b) The coach of the TEDU team finds out that the variance of the number of points scored is 25. Given this new information, construct a 95% two-sided confidence interval for the mean of the number of points scored by TEDU. c) Compare the confidence intervals found in part (a) and (b). Explain the reason behind the difference. Answer: Part (a) The table containing the mean, variance is shown below. X βX 9 β6 β3 β4 5 β3 1 β3 10 β6 X 74 59 62 61 70 62 66 62 75 59 Total ( X β X )2 81 36 9 16 25 9 1 9 100 36 322 The mean is 65. Use the equation π(π) = π(π) = 322 β(πβπΜ )2 πβ1 to find the variance. 9 = 35.78 Use the equation π. π· = βπ(π) to find the standard deviation. π. π· = β35.78 = 5.98 The t value corresponding to 95% confidence interval is 2.26. π2 Use the equation πΆ. πΌ = πΜ Β± π‘ to construct the confidence interval. πΆ. πΌ = 65 Β± 2.26 5.982 π 10 = 65 Β± 4.28 Therefore, the upper and lower limits are 65+4.28 and 65-4.28 Part (b) Given that, the variance of the distribution is 25. V(X)=25 π. π· = β25 =5 π2 Use the equation πΆ. πΌ = πΜ Β± π‘ to construct the interval. πΆ. πΌ = 65 Β± 2.26 Γ 52 π 10 πΆ. πΌ = 65 Β± 5.65 The upper and lower limits of the confidence interval is 65+5.65 and 65-5.65. Part (c) The standard deviation is defined as the degree of variance of value from mean value. In part (b) the standard deviation is higher compared to part (a). Therefore, the deviation is higher. So, the confidence interval got widened. Question #10 The scores of a certain population on the Wechsler Intelligence Scale for children (WISC) are thought to be normally distributed with a mean and standard deviation of 10. A simple random sample of twenty -five children from this population is taken and each is given the WISC. The mean of the twenty-five score is π₯Μ = 104.32. Based on these data a 95% confidence interval for \mu is computed. The 95% confidence interval for π is? Answer: Step 1 Given: π = 10 π₯Μ = 104.32 π = 25 Step 2 The 95% cocfidence interval for the mean is: π πΆ. πΌ. = π₯Μ Β± ππΌ/2 Γ βπ From the Z table ππΌ/2 = 1.96 = 104.32 Β± 1.96 Γ 10 β25 = 104.32 Β± 3.92 = (104.32 β 3.92, 104.32 + 3.92) = (100.40, 108.24) Answer: = (100.40, 108.24) Question #11 Mark each statement TRUE or FALSE. For these problems, assume that to estimate average time spent waiting at a restaurant, you sample 110 people and you get a 95% confidence interval of (4, 6) in minutes. 1. 95% of people wait between 4 and 6 minutes at the restaurant. 2. 95 of the people in your sample waited between 4 and 6 minutes. 3. You are 95% sure your sample resulted in a confidence interval the contains the true mean. Answer: Step 1 Given that there are 110 people in the sample and I get a 95% confidence interval of (4, 6) in minutes. This means I am 95% sure that my sample resulted in a confidence interval the contains the true mean of the population. This does not mean a range that contains 95% of the values. Step 2 Answer: 1. 95% of people wait between 4 and 6 minutes at the restaurant - False 2. 95 of the people in your sample waited between 4 and 6 minutes - False 3. You are 95% sure your sample resulted in a confidence interval the contains the true mean -True Question #12 In a science fair project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 358 trials, the touch therapists were correct 172 times. Complete parts (a) through (d).a) Given that Emily used a coin toss to select either her right hand or her left hand, what proportion of correct responses would be expected if the touch therapists made random guesses? (Type an integer or a decimal. Do not round.)b) Using Emily's sample results, what is the best point estimate of the therapists' success rate? (Round to three decimal places as needed.) c) Using Emily's sample results, construct a 90% confidence interval estimate of the proportion of correct responses made by touch therapists. Round to three decimal places as needed - ?
Questions and Answers #1 Confidence Intervals
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