Answer :
The 95% confidence interval is approximately (-1.8871, 1.5537), suggesting no significant difference in body temperature.
To calculate the 95% confidence interval estimate of the difference between the 8:00 am temperatures and the 12:00 am temperatures, we need to follow a few steps:
- Calculate the difference between each pair of temperatures:
8:00 am - 12:00 am
97.0 - 98.0 = -1.0
96.2 - 98.6 = -2.4
97.6 - 98.8 = -1.2
96.4 - 98.0 = -1.6
97.8 - 98.6 = -0.8
99.2 - 97.6 = 1.6
- Calculate the sample mean of the differences:
(-1.0 - 2.4 - 1.2 - 1.6 - 0.8 + 1.6) / 6 = -0.1667
- Calculate the sample standard deviation of the differences:
First, find the sum of squared differences from the mean:
[tex]((-1.0 + 0.1667)^2 + (-2.4 + 0.1667)^2 + (-1.2 + 0.1667)^2 + (-1.6 + 0.1667)^2 + (-0.8 + 0.1667)^2 + (1.6 + 0.1667)^2)[/tex] = 14.8002
Next, divide the sum by (n-1), where n is the number of differences:
14.8002 / (6-1) = 2.9600
Finally, take the square root to get the sample standard deviation:
sqrt(2.9600) ≈ 1.7204
- Calculate the margin of error (E):
The margin of error can be calculated using the formula:
E = t * (s / sqrt(n))
where t is the critical value from the t-distribution for a 95% confidence interval, s is the sample standard deviation, and n is the number of differences.
For a 95% confidence interval with 5 degrees of freedom (n-1), the critical value is approximately 2.571.
E = 2.571 * (1.7204 / sqrt(6)) ≈ 1.7204
- Construct the confidence interval:
The confidence interval estimate is given by:
(mean of the differences) ± (margin of error)
-0.1667 ± 1.7204
(-1.8871, 1.5537)
Therefore, the 95% confidence interval estimate of the difference between the 8:00 am temperatures and the 12:00 am temperatures is approximately (-1.8871, 1.5537).
Since this interval includes zero, we cannot conclude with 95% confidence that the body temperature is significantly different at both times.
However, it's important to note that this analysis is based on a small sample size and may not be representative of the population.
Learn more about Confidence interval
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