Questions: Each of the following is a confidence interval for μ= true average (i.e., population mean) resonance frequency (Hz) for all tennis rackets of a certain type: (117.6,118.4) (117.4,118.6) (a) What is the value of the sample mean resonance frequency? Hz (b) Both intervals were calculated from the same sample data. The confidence level for one of these intervals is 90% and for the other is 99%. Which of the intervals has the 90% confidence level, and why? The first interval has the 90% confidence level because it is a narrower interval. The second interval has the 90% confidence level because it is a wider interval. The second interval has the 90% confidence level because it is a narrower interval. The first interval has the 90% confidence level because it is a wider interval.

Each of the following is a confidence interval for μ= true average (i.e., population mean) resonance frequency (Hz) for all tennis rackets of a certain type:
(117.6,118.4)
(117.4,118.6)
(a) What is the value of the sample mean resonance frequency?
Hz
(b) Both intervals were calculated from the same sample data. The confidence level for one of these intervals is 90% and for the other is 99%. Which of the intervals has the 90% confidence level, and why?
The first interval has the 90% confidence level because it is a narrower interval.
The second interval has the 90% confidence level because it is a wider interval.
The second interval has the 90% confidence level because it is a narrower interval.
The first interval has the 90% confidence level because it is a wider interval.

Transcript text: Each of the following is a confidence interval for $\mu=$ true average (i.e., population mean) resonance frequency $(\mathrm{Hz})$ for all tennis rackets of a certain type: $(117.6,118.4)$ $(117.4,118.6)$ (a) What is the value of the sample mean resonance frequency? $\qquad$ Hz (b) Both intervals were calculated from the same sample data. The confidence level for one of these intervals is $90 \%$ and for the other is $99 \%$. Which of the intervals has the $90 \%$ confidence level, and why? The first interval has the $90 \%$ confidence level because it is a narrower interval. The second interval has the $90 \%$ confidence level because it is a wider interval. The second interval has the $90 \%$ confidence level because it is a narrower interval. The first interval has the $90 \%$ confidence level because it is a wider interval.

Solution

Solution Steps

To solve this problem, we need to determine the sample mean resonance frequency and identify which confidence interval corresponds to the 90% confidence level.

(a) The sample mean is the midpoint of the confidence interval. We can calculate it by averaging the lower and upper bounds of either interval.

(b) The confidence interval with the narrower range corresponds to the 90% confidence level because a higher confidence level (e.g., 99%) requires a wider interval to ensure the true mean is captured.

Step 1: Calculate the Sample Mean Resonance Frequency

To find the sample mean resonance frequency, we take the midpoint of either confidence interval. For the interval $(117.6, 118.4)$, the sample mean is calculated as follows:

\[ \text{Sample Mean} = \frac{117.6 + 118.4}{2} = 118.0 \]

Similarly, for the interval $(117.4, 118.6)$, the sample mean is:

\[ \text{Sample Mean} = \frac{117.4 + 118.6}{2} = 118.0 \]

Since both intervals are derived from the same sample data, the sample mean is the same for both, which is $118.0$.

Step 2: Determine the Confidence Level of Each Interval

The width of a confidence interval is the difference between its upper and lower bounds. For the interval $(117.6, 118.4)$, the width is:

\[ \text{Width}_1 = 118.4 - 117.6 = 0.8 \]

For the interval $(117.4, 118.6)$, the width is:

\[ \text{Width}_2 = 118.6 - 117.4 = 1.2 \]

A narrower interval corresponds to a lower confidence level. Therefore, the interval $(117.6, 118.4)$ with a width of $0.8$ is the narrower one and corresponds to the $90\%$ confidence level.