To estimate the mean wrist extension for people using the new mouse design, we can construct a 90% confidence interval based on the data of wrist extensions measured from 24 students. Here are the detailed steps for calculating the 90% confidence interval:
1. Data Collection:
The wrist extension measurements for the 24 students are as follows:
[tex]\[
26, 28, 23, 26, 27, 25, 25, 24, 24, 24, 25, 28, 22, 25, 24, 28, 27, 26, 31, 25, 28, 27, 27, 25
\][/tex]
2. Sample Size (n):
The total number of students (data points) is [tex]\( n = 24 \)[/tex].
3. Sample Mean ([tex]\(\bar{x}\)[/tex]):
The sample mean is the average of the wrist extension measurements.
[tex]\[
\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{26 + 28 + 23 + \dots + 25}{24}
\][/tex]
4. Sample Standard Deviation (s):
The sample standard deviation is calculated using the formula:
[tex]\[
s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}
\][/tex]
Here, the degrees of freedom is [tex]\( n-1 = 23 \)[/tex].
5. Confidence Level:
The confidence level is 90%. For a 90% confidence interval, we need to find the z-score that corresponds to the middle 90% of the standard normal distribution. This z-score (critical value) is approximately 1.645.
6. Margin of Error (E):
The margin of error is calculated using the formula:
[tex]\[
E = z \left( \frac{s}{\sqrt{n}} \right)
\][/tex]
where [tex]\( z \approx 1.645 \)[/tex].
7. Confidence Interval Calculation:
The confidence interval for the population mean [tex]\(\mu\)[/tex] is given by:
[tex]\[
\text{Lower Bound} = \bar{x} - E
\][/tex]
[tex]\[
\text{Upper Bound} = \bar{x} + E
\][/tex]
8. Final Result:
Based on the data provided, the 90% confidence interval for the mean wrist extension is (rounded to three decimal places):
[tex]\[
(25.164, 26.502)
\][/tex]
Thus, we can be 90% confident that the true mean wrist extension for people using this new mouse design lies between 25.164 and 26.502 degrees.
