Answer:
Step-by-step explanation:
We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean
For the null hypothesis,
µ = 82
For the alternative hypothesis,
µ ≠ 82
This is a 2 tailed test
Since the sample mean and sample standard deviation is given, the t test would be used to determine the test statistic. The formula is
t = (x - µ)/(s/√n)
Where
x = sample mean = 87
µ = population mean = 82
s = samples standard deviation = 10
n = 25
t = (87 - 82)/(10/√25) = 2.5
α = 1 - Confidence level
α = 1 - 0.95 = 0.05
Since α = 0.05, the critical value is determined from the t distribution table.
For the left, α/2 = 0.05/2 = 0.025
For the right of 0.025 = 1 - 0.025 = 0.975
To determine the t score from the t distribution table, we would find the degree of freedom, df and look for the corresponding α value.
df = n - 1 = 25 - 1 = 24
t score = critical value = ±2.064
In order to reject the null hypothesis, the test statistic must be smaller than - 2.5 or greater than 2.5
Since - 2.064 > - 2.5 and 2.064 < 2.5, we would fail to reject the null hypothesis.
Confidence interval is written in the form,
(Sample mean - margin of error, sample mean + margin of error)
Margin of error = z × s/√n
Where
s = sample standard deviation
z = t score
Margin of error = 2.064 × 10/√25
= 4.13
Confidence interval = 87 ± 4.13
the lower limit of this confidence interval is
87 - 4.13 = 82.87
the upper limit of this confidence interval is
87 + 4.13 = 91.13
