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------------------------------------------------ Assume that a sample is used to estimate a population mean [tex]\mu[/tex]. Find the 99.9% confidence interval for a sample of size 65 with a mean of 37.9 and a standard deviation of 15.4. Enter your answer as an open interval (i.e., parentheses) accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

99.9% C.I. = ____________

The 99.9% confidence interval for the population mean μ, based on a sample of size 65 with a mean of 37.9 and a standard deviation of 15.4, is (31.6, 44.2).

To calculate the 99.9% confidence interval for the population mean, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

First, let's calculate the critical value. Since the sample size is 65, we need to find the z-score associated with a 99.9% confidence level. We can use a standard normal distribution table or a calculator to find this value.

For a 99.9% confidence level, the corresponding z-score is approximately 3.29.

Next, we need to calculate the standard error, which represents the standard deviation of the sampling distribution of the sample mean. The standard error can be calculated using the formula:

Standard Error = Standard Deviation / √(Sample Size)

Given that the sample size is 65 and the standard deviation is 15.4, we can substitute these values into the formula:

Standard Error = 15.4 / √(65) ≈ 1.909

Now we can plug in the values into the confidence interval formula:

Confidence Interval = 37.9 ± (3.29 * 1.909)

Calculating the values within the parentheses:

Confidence Interval = 37.9 ± 6.28

Finally, we can express the confidence interval as an open-interval with one decimal place accuracy:

99.9% C.I. = (31.6, 44.2)

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roniksingha96 roniksingha96 · Answer 2 · 2025-02-07T03:29:49-05:00

The 99.9% confidence interval for the population mean μ, based on a sample of size 65 with a mean of 37.9 and a standard deviation of 15.4, is (31.6, 44.2).

To calculate the 99.9% confidence interval for the population mean, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

First, let's calculate the critical value. Since the sample size is 65, we need to find the z-score associated with a 99.9% confidence level. We can use a standard normal distribution table or a calculator to find this value.

For a 99.9% confidence level, the corresponding z-score is approximately 3.29.

Next, we need to calculate the standard error, which represents the standard deviation of the sampling distribution of the sample mean. The standard error can be calculated using the formula:

Standard Error = Standard Deviation / √(Sample Size)

Given that the sample size is 65 and the standard deviation is 15.4, we can substitute these values into the formula:

Standard Error = 15.4 / √(65) ≈ 1.909

Now we can plug in the values into the confidence interval formula:

Confidence Interval = 37.9 ± (3.29 * 1.909)

Calculating the values within the parentheses:

Confidence Interval = 37.9 ± 6.28

Finally, we can express the confidence interval as an open interval with one decimal place accuracy:

99.9% C.I. = (31.6, 44.2)

Learn more about confidence interval here:

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