1 - Graded Week 5 Homework-Mark Kasule

Student: Mark Kasule Date: 03/19/24 Instructor: Petal Sumner Course: STAT 200 6961 Introduction to Statistics (1) Assignment: Go - Graded Week 5 Homework Use the sample data and confidence level given below to complete parts (a) through (d). A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, and who said "yes." Use a confidence level. n = 1111 x = 510 95% Click the icon to view a table of z-scores. 1 a) Find the best point estimate of the population proportion p. The sample proportion is the best point estimate of the population proportion p. p Find the sample proportion , rounding to three decimal places. p p = x n = 510 1111 = 0.459 b) Identify the value of the margin of error E. The margin of error, E, for a population proportion is found using the following formula, where is the critical value separating an area of in the right tail of the standard normal distribution, is the sample proportion, and n is the sample size. z α / 2 α / 2 p = 1 − , q p E = z α / 2 p q n Because a % confidence interval is requested, . Use the accompanying table of z scores to find the critical value, . Recall that is the z score separating an area of in the right tail of the standard normal distribution. 95 α = 0.05 z α / 2 z α / 2 α / 2 z α / 2 = z 0.05 / 2 = z 0.025 = 1.96 Now calculate the margin of error E, rounding to three decimal places, using the formula given above. Substitute and z = 1.96, α / 2 = 0.459, p = 1 − 0.459, q n = 1111. E = z α / 2 p q n = 1.96 0.459(1 − 0.459) 1111 = 0.029 c) Construct the confidence interval. First check that the requirements to construct a confidence interval used to estimate a population proportion are met. The requirements are shown below. 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied. 3. There are at least 5 successes and at least 5 failures. Assume that the polling methods used by the research institute result in simple random samples. All three requirements are met. The polling methods used by the research institute result in simple random samples; the conditions for a binomial experiment are satisfied because there is a fixed number of independent trials, there are two categories of outcomes, and the probability remains constant; and there are at least 5 people who said "yes" and 5 people who said "no." Go - Graded Week 5 Homework-Mark Kasule https://tdx.acs.pearsonprd.tech/api/v1/print/highered 1 of 4 3/19/24, 2:09 AM

1: Standard Normal (z) Distribution The confidence interval is given by the following formula, where is the sample proportion, E is the margin of error, and and are the lower and upper limits, respectively. p − E p + E p − E < p < + E p p First calculate the lower limit, , substituting and E . − E p p = 0.459 = 0.029 − E p = 0.459 − 0.029 = 0.430 Next calculate the upper limit, . + E p + E p = 0.459 + 0.029 = 0.488 Therefore, the confidence interval is 0.430 < p < 0.488. d) Write a statement that correctly interprets the confidence interval. The confidence level is the probability 1 (in this case , or %) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. In this case, if one were to select many different samples of size and construct the corresponding confidence intervals, % of them would actually contain the value of the population proportion p. − α 0.95 95 1111 95 One must be careful to interpret confidence interval levels correctly. There is a correct interpretation and many different and creative incorrect interpretations. Use this information to write a statement that correctly interprets the confidence interval. Go - Graded Week 5 Homework-Mark Kasule https://tdx.acs.pearsonprd.tech/api/v1/print/highered 2 of 4 3/19/24, 2:09 AM

Go - Graded Week 5 Homework-Mark Kasule https://tdx.acs.pearsonprd.tech/api/v1/print/highered 4 of 4 3/19/24, 2:09 AM