Hypotheses Testing

Question

Before Covid, a census of the Region stated that the average salary of working adults was $50,000. I want to see if this number has changed. I go out and randomly choose 40 people from the area. I find that they have an average salary of $46,000, with a standard deviation of $22,000. Conduct a hypothesis test at the 10% level of significance to see if the average Salary has changed.

State the null and alternate hypotheses.

State the correct tcv or zcv score as applicable.

Calculate the test score and state the conclusion

Answer 1

Hi Chad,

Null hypothesis always means "nothing," as in "no difference, so:

H0: mu = 50000

HA: mu not equal to 50000

Note that there was no greater than or less than direction given in the problem, so you have to use the two-sided, not equal to alternative.

You have only a sample standard deviation here, no sigma, so you have to use t. In terms of critical value, we go to the t-table, look at the bottom row for 90% confidence (1 - significance level). You also need degrees of freedom:

df = n - 1 = 40 -1 = 39

No row for 39 degrees of freedom, so you have to go with next closest without going over, so that's 30. Thus:

t* = 1.697

As for the test statistic:

t= (xbar - mu) / SE

xbar= sample mean = 46000

mu = 50000

SE = standard error = s/sqrt(n) = 22000/sqrt (40) = 3478.51

t = (46000 - 50000) / 3478.51

t = -1.150

Using the critical value method, if the absolute value of your t-test statistic exceeds the critical, you reject the null in favor of the alternative hypothesis. In this case, though:

abs(- 1.150) < 1.697, so:

Fail to reject H0 and conclude true population mean salary does not differ significantly from $50000.

I hope this helps.