A simple random sample of 90 is drawn from a normally distributed population, and the mean is found to be 138, with a standard deviation of 34. What is the 90% confidence interval for the population mean? Use the table below to help you answer the question.

Question

A simple random sample of 90 is drawn from a normally distributed population, and the mean is found to be 138, with a standard deviation of 34. What is the 90% confidence interval for the population mean? Use the table below to help you answer the question.

Confidence Level
90%
95%
99%
z*-score
1.645
1.96
2.58

Answer 1

132.10 to 143.90

Answer 2

90% = mean ± 1.645 SEm

SEm = SD/√n

Answer 3

To find the 90% confidence interval for the population mean, we can use the formula:

Confidence interval = sample mean +/- (z*-score) * (standard deviation / sqrt(sample size))

Given the information:

Sample mean (x̄) = 138
Standard deviation (σ) = 34
Sample size (n) = 90
z*-score for 90% confidence level = 1.645

Substituting these values into the formula:

Confidence interval = 138 +/- (1.645) * (34 / sqrt(90))

Calculating the standard error (standard deviation / sqrt(sample size)):

Standard error = 34 / sqrt(90)

Now, we can calculate the confidence interval:

Confidence interval = 138 +/- (1.645) * (standard error)

Confidence interval = 138 +/- (1.645) * (34 / sqrt(90))

Confidence interval = 138 +/- (1.645) * (34 / 9.4868)

Confidence interval = 138 +/- (1.645) * (3.5854)

Confidence interval = 138 +/- 5.9014

Therefore, the 90% confidence interval for the population mean is (132.0986, 143.9014).

Answer 4

To find the 90% confidence interval for the population mean, we can use the formula:

Confidence Interval = sample mean ± (z * (standard deviation / square root of sample size))

Given that the sample mean is 138, the standard deviation is 34, and the sample size is 90, we need to determine the value of z for a 90% confidence level.

Looking at the table provided, the z-score for a 90% confidence level is 1.645.

Substituting the values into the formula, we get:

Confidence Interval = 138 ± (1.645 * (34 / square root of 90))

To calculate the square root of 90, we find that it is approximately 9.49.

Substituting this value, we get:

Confidence Interval = 138 ± (1.645 * (34 / 9.49))

Now, we can compute the expression inside the parentheses:

34 / 9.49 ≈ 3.58

Substituting this value again, we have:

Confidence Interval = 138 ± (1.645 * 3.58)

To simplify further:

Confidence Interval = 138 ± 5.8951

Now we can calculate the upper and lower bounds of the confidence interval:

Upper Bound = 138 + 5.8951 = 143.8951
Lower Bound = 138 - 5.8951 = 132.1049

Therefore, the 90% confidence interval for the population mean is approximately (132.1049, 143.8951).