Question
For a group of 10 men subjected to a stress situation, the mean number of heartbeats per minute was 126 and the standard deviation was 4. Find the 95% confidence interval of the true mean. Populations seems to be normally distributed.
Please round to the nearest whole number and write your answer in the form ( , ); for example (31,34) or (25,46). Do not type in any extra spaces.
Please round to the nearest whole number and write your answer in the form ( , ); for example (31,34) or (25,46). Do not type in any extra spaces.
Answers
GPT-4o mini
To find the 95% confidence interval for the true mean, we use the formula:
\[
\text{Confidence Interval} = \bar{x} \pm t^* \left(\frac{s}{\sqrt{n}}\right)
\]
where:
- \(\bar{x}\) is the sample mean,
- \(t^*\) is the t-score for the desired confidence level and degrees of freedom,
- \(s\) is the sample standard deviation,
- \(n\) is the sample size.
Given:
- \(\bar{x} = 126\)
- \(s = 4\)
- \(n = 10\)
First, we need to find the t-score corresponding to a 95% confidence level and 9 degrees of freedom (since \(n - 1 = 10 - 1 = 9\)).
Using a t-table or calculator, we find that \(t^*\) for 9 degrees of freedom at a 95% confidence level is approximately 2.262.
Next, we calculate the standard error (SE):
\[
SE = \frac{s}{\sqrt{n}} = \frac{4}{\sqrt{10}} \approx \frac{4}{3.162} \approx 1.264
\]
Now we can calculate the margin of error (ME):
\[
ME = t^* \cdot SE = 2.262 \cdot 1.264 \approx 2.861
\]
Finally, we can calculate the confidence interval:
\[
\text{Lower limit} = \bar{x} - ME = 126 - 2.861 \approx 123
\]
\[
\text{Upper limit} = \bar{x} + ME = 126 + 2.861 \approx 129
\]
Rounding to the nearest whole number, the 95% confidence interval for the true mean is:
\((123,129)\)
\[
\text{Confidence Interval} = \bar{x} \pm t^* \left(\frac{s}{\sqrt{n}}\right)
\]
where:
- \(\bar{x}\) is the sample mean,
- \(t^*\) is the t-score for the desired confidence level and degrees of freedom,
- \(s\) is the sample standard deviation,
- \(n\) is the sample size.
Given:
- \(\bar{x} = 126\)
- \(s = 4\)
- \(n = 10\)
First, we need to find the t-score corresponding to a 95% confidence level and 9 degrees of freedom (since \(n - 1 = 10 - 1 = 9\)).
Using a t-table or calculator, we find that \(t^*\) for 9 degrees of freedom at a 95% confidence level is approximately 2.262.
Next, we calculate the standard error (SE):
\[
SE = \frac{s}{\sqrt{n}} = \frac{4}{\sqrt{10}} \approx \frac{4}{3.162} \approx 1.264
\]
Now we can calculate the margin of error (ME):
\[
ME = t^* \cdot SE = 2.262 \cdot 1.264 \approx 2.861
\]
Finally, we can calculate the confidence interval:
\[
\text{Lower limit} = \bar{x} - ME = 126 - 2.861 \approx 123
\]
\[
\text{Upper limit} = \bar{x} + ME = 126 + 2.861 \approx 129
\]
Rounding to the nearest whole number, the 95% confidence interval for the true mean is:
\((123,129)\)