Question

You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 60 dates, the mean record high daily temperature in a certain city has a mean of 83.46∘F. Assume the population standard deviation is 15.33∘F.

Answer 1

To construct confidence intervals for the population mean, we use known population standard deviation and the formula: (sample mean - margin of error, sample mean + margin of error). For a 90% confidence interval, the margin of error is calculated using the critical value of approximately 1.645, and for a 95% confidence interval, the critical value is approximately 1.96. The confidence intervals for the given sample mean of 83.46°F and population standard deviation of 15.33°F are (81.82, 85.10) for the 90% interval and (81.43, 85.49) for the 95% interval. The 95% confidence interval is wider, as it accounts for a larger range of values.

Step-by-step explanation:

When constructing confidence intervals for the population mean using a known population standard deviation, we use the formula:

(sample mean - margin of error, sample mean + margin of error)

To calculate the margin of error, we need to determine the critical value corresponding to the desired confidence level. For the 90% confidence interval, the critical value is approximately 1.645, and for the 95% confidence interval, the critical value is approximately 1.96.

Given a sample mean of 83.46°F and a population standard deviation of 15.33°F, we can calculate the confidence intervals as follows:

For the 90% confidence interval:

(83.46 - (1.645 * (15.33 / √60)), 83.46 + (1.645 * (15.33 / √60)))

(81.82, 85.10)

For the 95% confidence interval:

(83.46 - (1.96 * (15.33 / √60)), 83.46 + (1.96 * (15.33 / √60)))

(81.43, 85.49)

Interpreting the results:

The 90% confidence interval suggests that we are 90% confident that the true population mean falls within the range of 81.82°F to 85.10°F. Similarly, the 95% confidence interval indicates that we are 95% confident the true population mean lies within the range of 81.43°F to 85.49°F.

Comparing the widths of the confidence intervals:

The 95% confidence interval is wider than the 90% confidence interval. This is because a higher confidence level requires accounting for a wider range of values, resulting in a broader interval.

Answer 2

To create 90% and 95% confidence intervals for a population mean with a known standard deviation, we calculate the error bound for the mean (EBM) using the formula EBM = Z * (sigma/√n) with respective z-scores for each confidence level. The 95% confidence interval will be wider than the 90% due to a higher z-score reflecting increased certainty.

Step-by-step explanation:

When constructing confidence intervals for the population mean with a known population standard deviation, you use the normal distribution. Given a sample mean (µ) of 83.46°F and a population standard deviation (sigma) of 15.33°F from a sample of 60 dates, we can calculate the 90% and 95% confidence intervals for the population mean.

Calculating the Error Bound for the Mean (EBM)

The error bound for the population mean (EBM) depends on the confidence level selected and is calculated using the formula EBM = Z * (sigma/√n), where Z is the z-score that corresponds to the desired confidence level and n is the sample size.

90% Confidence Interval

The z-score corresponding to a 90% confidence level is approximately 1.645 since it captures the central 90% of the normal distribution. Calculating the EBM:
EBM = 1.645 * (15.33/√60). This gives us an EBM which we can use to find the interval: (83.46 - EBM, 83.46 + EBM).

95% Confidence Interval

For a 95% confidence level, the z-score is approximately 1.96. Using the same formula: EBM = 1.96 * (15.33/√60), you find the corresponding EBM and calculate the interval: (83.46 - EBM, 83.46 + EBM).

Comparing the widths of the confidence intervals, the 95% interval will be wider than the 90% because the z-score is higher for 95% confidence, reflecting that we require more certainty that the interval will contain the true population mean.