Question

A random sample of the average debt (in dollars) at graduation from 18 of the top 100 public colleges and universities is listed below. What is the required conditions for the statistical test in part (a)?

A random sample of the average debt (in dollars) at graduation from 18 of the top 100 public colleges and universities is listed below. 16012 15784 16597 18105 12665 14734 17225 16953 15309 15297 14437 14835 13607 13374 19410 18385 22312 16656 a) Is there sufficient evidence at the 10% significance level to conclude that the population mean debt at graduation is less than $18000? b) What is the required conditions for the statistical test in part (a)?

Answer 1

Null hypothesis H0: $\mu$ = $18000. Population mean debt at graduation is equal to $18000.

Alternative hypothesis Ha: $\mu$ < $18000. Population mean debt at graduation is less than $18000.

From the data, Sample mean $\bar{x}$ = 16205.39

Sample standard deviation, SD = 2353.857

Standard error of sample mean, SE = SD / $\sqrt{n}$ = 2353.857 / $\sqrt{18}$ = 554.8094

Test statistic, t = ($\bar{x}$ - $\mu$ ) / SE = (16205.39 - 18000) / 554.8094 = -3.23

Degree of freedom, df = n-1 = 18 - 1 = 17

P-value = P(t < -3.23, df = 17) = 0.0025

Since p-value is less than 0.1 significance level, we reject null hypothesis H0 and conclude that there is significant evidence that
population mean debt at graduation is less than $18000.

b)

The required conditions for the statistical test (one-sample t test) in part (a) are -

• The sampling method is simple random sampling.
• The sampling distribution is normal or nearly normal.