# Hypothesis Testing & Confidence Interval: Wages

Assume that the average national wage for all educational levels is $15 with a standard deviation of $4. We want to find out whether individuals with higher education tend to make higher wages than on the average nationally for all educational levels. Our null hypothesis is that individuals with an educational level higher than high school do not make higher wages than the average wage for all Americans with any educational level. Our alternative hypothesis is that individuals with an education higher than high school do make higher wages than the wage of an average American. On a new tab in the Excel spreadsheet, list the wages for all observations in which education was higher than 12 (higher than a high school degree). Calculate the mean and the standard error of the mean for wages for this sample. Your standard error of the mean should use the unbiased standard deviation (you can use either the population standard deviation, which is by nature unbiased, or you can calculate the unbiased standard deviation of the sample using formula 2.5).

1.) Calculate the Z score for the wage sample you constructed in #2 comparing the sample means to the national mean ("z score for groups").

2.)Find the probability that a random sample of persons in the U.S. could have a mean of wages as high as the mean of the wages in your sample of higher educated individuals (use the Z score from #3 to find the area beyond Z).

3.)Is the null hypothesis rejected at a 0.05 significance level?

4.)Calculate a 95% confidence interval CI for the mean wage for your sample using an unbiased standard deviation.

5.)Interpret the CI

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Probability, Sampling Distribution, and Tools of Inference

Assume that the average national wage for all educational levels is $15 with a standard deviation of $4. We want to find out whether individuals with higher education tend to make higher wages than on the average nationally for all educational levels. Our null hypothesis is that individuals with an educational level higher than high school do not make higher wages than the average wage for all Americans with any educational level. Our alternative hypothesis is that individuals with an education higher than high school do make higher wages than the wage of an average American. On a new tab in the Excel spreadsheet, list the wages for all observations in which education was higher than 12 (higher than a high school degree). Calculate the mean and the standard error of the mean for wages for this sample. Your standard error of the mean should use the unbiased standard deviation (you can use either the population standard deviation, which is by nature unbiased, or you can calculate the unbiased standard deviation of the sample using formula 2.5).

Answers

Mean = = 18.33333333

Given that population standard deviation, σ = 4

Therefore, standard error, Se = σ/√n = 4/√15 = 1.032795559

1.) Calculate the Z score for the wage sample you constructed in #2 comparing the sample means to the national mean ("z score for groups").

Z score, , where =18.33333333, µ = 15, n = 15, Se = 1.032795559

Therefore, = 3.22749

2.) Find the probability that a random sample of persons in the U.S. could have a mean of wages as high as the mean of the wages in your sample of higher educated individuals (use the Z score from #3 to find the area beyond Z).

P (Z > 3.22749) = 0.00062

3.) Is the null hypothesis rejected at a 0.05 significance level?

Yes, the null hypothesis is rejected, since the p-value (0.00062) is less than the significance level 0.05. The sample provides sufficient evidence to conclude that individuals with an education higher than high school do make higher wages than the wage of an average American.

Details

Z Test of Hypothesis for the Mean

Data

Null Hypothesis = 15

Level of Significance 0.05

Population Standard Deviation 4

Sample Size 15

Sample Mean 18.33333333

Intermediate Calculations

Standard Error of the Mean 1.032795559

Z Test Statistic 3.227486122

Upper-Tail Test

Upper Critical Value 1.644853627

p-Value 0.000624415

Reject the null hypothesis

4.) Calculate a 95% confidence interval CI for the mean wage for your sample using an unbiased standard deviation.

The confidence interval is given by , where = 18.33333333, Se = 1.032795559, n = 15, = 1.959963985

That is,

= (16.31, 20.36)

Details

Confidence Interval Estimate for the Mean

Data

Population Standard Deviation 4

Sample Mean 18.33333333

Sample Size 15

Confidence Level 95%

Intermediate Calculations

Standard Error of the Mean 1.032795559

Z Value -1.959963985

Interval Half Width 2.024242099

Confidence Interval

Interval Lower Limit 16.31

Interval Upper Limit 20.36

5.) Interpret the CI

With 95% confidence we can claim that population mean wages of individuals with an education higher than high school is within ($16.31, $20.36).

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