Find a confidence interval for ƒÊ assuming that each sample is from a normal population.
a. �P x = 14, ƒÐ = 4, n = 5, 90 percent confidence
b. �P x = 37, ƒÐ = 5, n = 15, 99 percent confidence
c. �P x = 121, ƒÐ = 15, n = 25, 95 percent confidence
Need to show my work/formula
Thanks,
Assuming two-tailed test, x = mean, ƒD = Standard deviation:
90% = mean ± 1.645 SEm
SEm = SD/√n = Standard Error of the mean
99% = mean ± 2.575 SEm
95% = mean ± 1.96 SEm
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions and their related Z scores.
To find the confidence interval for µ (population mean), you can use the following formula:
Confidence Interval = x ± (Z * σ/√n)
Where:
- x is the sample mean
- Z is the critical value from the standard normal distribution corresponding to the desired confidence level
- σ is the population standard deviation
- n is the sample size
Now, let's calculate the confidence intervals for each scenario:
a. x = 14, σ = 4, n = 5, 90 percent confidence
First, find the critical value corresponding to a 90 percent confidence level. This value can be found in the Z-table or using a calculator. The critical value for a 90 percent confidence level is approximately 1.645.
Confidence Interval = 14 ± (1.645 * 4/√5)
Confidence Interval ≈ 14 ± (1.645 * 4/2.236)
Confidence Interval ≈ 14 ± (1.645 * 1.792)
Confidence Interval ≈ 14 ± 2.948
Confidence Interval ≈ (14 - 2.948, 14 + 2.948)
Confidence Interval ≈ (11.052, 16.948)
Therefore, the confidence interval for µ is (11.052, 16.948) at a 90 percent confidence level.
b. x = 37, σ = 5, n = 15, 99 percent confidence
The critical value for a 99 percent confidence level is approximately 2.576.
Confidence Interval = 37 ± (2.576 * 5/√15)
Confidence Interval ≈ 37 ± (2.576 * 5/√15)
Confidence Interval ≈ 37 ± (2.576 * 5/3.873)
Confidence Interval ≈ 37 ± (2.576 * 1.29)
Confidence Interval ≈ 37 ± 3.325
Confidence Interval ≈ (37 - 3.325, 37 + 3.325)
Confidence Interval ≈ (33.675, 40.325)
Therefore, the confidence interval for µ is (33.675, 40.325) at a 99 percent confidence level.
c. x = 121, σ = 15, n = 25, 95 percent confidence
The critical value for a 95 percent confidence level is approximately 1.96.
Confidence Interval = 121 ± (1.96 * 15/√25)
Confidence Interval ≈ 121 ± (1.96 * 15/5)
Confidence Interval ≈ 121 ± (1.96 * 3)
Confidence Interval ≈ 121 ± 5.88
Confidence Interval ≈ (121 - 5.88, 121 + 5.88)
Confidence Interval ≈ (115.12, 126.88)
Therefore, the confidence interval for µ is (115.12, 126.88) at a 95 percent confidence level.