Answer :
The 95% confidence interval for the mean high temperature of towns is approximately (97.66, 99.50)
How to calculate confidence interval
To find the 95% confidence interval of the mean high temperature of towns based on the given data, use the formula below
[tex]CI = \bar{x} \± (t * (s/\sqrt n))[/tex]
Where:
CI represents the confidence interval
[tex]\bar{x}[/tex] is the sample mean
t is the critical value from the t-distribution for the desired confidence level and degrees of freedom
s is the sample standard deviation
n is the sample size
Given data:
Sample size (n) = 6
Sample mean ([tex]\bar{x}[/tex]) = (99 + 97.5 + 97.9 + 99.4 + 97 + 97.7) / 6 ≈ 98.58 (rounded to two decimal places)
To calculate the sample standard deviation (s),
compute the differences between each temperature value and the sample mean,
square those differences,
sum them up,
divide by (n-1), and
take the square root:
[tex]s= \sqrt((99-98.58)^2 + (97.5-98.58)^2 + (97.9-98.58)^2 + (99.4-98.58)^2 + (97-98.58)^2 + (97.7-98.58)^2)/(6-1)][/tex]
s ≈ 0.877
Next, determine the critical value (t) for a 95% confidence level with 5 degrees of freedom (n-1).
Using a t-table, t ≈ 2.571
Now plug in the values into the confidence interval formula:
CI = 98.58 ± (2.571 * (0.877/√6))
CI ≈ 98.58 ± (2.571 * 0.358)
CI ≈ 98.58 ± 0.921
Therefore, the 95% confidence interval for the mean high temperature of towns is approximately (97.66, 99.50) (rounded to two decimal places).
