A simple random sample of 60 items resulted in a sample mean of 90. The population standard deviation is 13.a. Compute the 95% confidence interval for the population mean (to 1 decimal).( , ) I got 93.2 and 86.7, but I keep getting it wrong&nbsp;b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean (to 2 decimals).( , )c. What is the effect of a larger sample size on the margin of error?&nbsp;

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A simple random sample of 60 items resulted in a sample mean of 90. The population standard deviation is 13.a. Compute the 95% confidence interval for the population mean (to 1 decimal).( , ) I got 93.2 and 86.7, but I keep getting it wrong&amp;nbsp;b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean (to 2 decimals).( , )c. What is the effect of a larger sample size on the margin of error?&amp;nbsp;

Jazz Frenia · Accepted Answer

a. To compute the 95% confidence interval for the population mean, we can use the formula:(x¯−zσn,x¯+zσn)where:x¯ is the sample mean,σ is the population standard deviation,n is the sample size,z is the critical value corresponding to the desired confidence level.Given:x¯=90,σ=13,n=60.To find the critical value, we can use a standard normal distribution table or a statistical software. For a 95% confidence level, the critical value is approximately 1.96.Plugging in the values, we get:(90−1.961360,90+1.961360)Simplifying the expression, we find:(86.72,93.28)Therefore, the 95% confidence interval for the population mean is (86.72,93.28).b. Assuming the same sample mean of 90 and a larger sample size of 120, we can compute the 95% confidence interval using the same formula:(x¯−zσn,x¯+zσn)Given:x¯=90,σ=13,n=120.Using the critical value of 1.96 for a 95% confidence level, we have:(90−1.9613120,90+1.9613120)Simplifying the expression, we find:(88.16,91.84)Therefore, the 95% confidence interval for the population mean with a sample size of 120 is (88.16,91.84).c. The effect of a larger sample size on the margin of error is that it reduces the margin of error. As the sample size increases, the standard error decreases, resulting in a narrower confidence interval. A larger sample size provides more information about the population, leading to increased precision in estimating the population mean.

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