A random sample of 90 observations produced a mean that = 25.9 and a standard deviation s =2.4
(a)Find a 95% confidence interval for u
(b)Find a 90% confidence interval for u
(c)Find a 99% confidence interval for u
(Use integers or decimals for any number in the expression. Round to two decimal places
a. 25.4 -+ 1.96 * 2.4 /sqrt((90))
b. 25.4 -+ 1.645 * 2.4 /sqrt((90))
c. 25.4 -+ 2.575 * 2.4 /sqrt((90))
To find the confidence intervals, we will use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / square root of sample size)
(a) 95% confidence interval for 𝜇:
For a 95% confidence interval, we need to find the critical value first. The critical value can be found using a t-distribution table or a statistical calculator with the degree of freedom (df) = sample size - 1.
Since the sample size is 90, the df = 90 - 1 = 89.
Using the critical value for a 95% confidence level and df = 89, we can find the value to be 1.984 (rounded to three decimal places).
Now, let's calculate the confidence interval:
Confidence Interval = 25.9 ± (1.984) * (2.4 / √90)
Confidence Interval = 25.9 ± 0.462
Confidence Interval = [25.438, 26.362]
Therefore, the 95% confidence interval for 𝜇 is [25.438, 26.362].
(b) 90% confidence interval for 𝜇:
Using the same process as in part (a), we determine that the critical value for a 90% confidence level and df = 89 is 1.663 (rounded to three decimal places).
Confidence Interval = 25.9 ± (1.663) * (2.4 / √90)
Confidence Interval = 25.9 ± 0.389
Confidence Interval = [25.511, 26.289]
The 90% confidence interval for 𝜇 is [25.511, 26.289].
(c) 99% confidence interval for 𝜇:
The critical value for a 99% confidence level and df = 89 is 2.626 (rounded to three decimal places).
Confidence Interval = 25.9 ± (2.626) * (2.4 / √90)
Confidence Interval = 25.9 ± 0.588
Confidence Interval = [25.312, 26.488]
The 99% confidence interval for 𝜇 is [25.312, 26.488].