A tax specialist knows that tax refunds for people who use a firm to do their tax returns are normally distributed with a mean of $150 and a standard deviation of
$43. The specialist believes that the company, Taxco, uses under trained staff and gets a lower average tax refund for its customers. He takes a random sample of 6 customers from Taxco’s records and finds the following refunds:
Refund($)144, 38, 130, 160, 81, 135
Run a full hypothesis test at the 5% level of significance to test the tax specialists belief about Taxco.
To run a hypothesis test, we need to set up our null and alternative hypotheses.
Null hypothesis (H0): The average tax refund for Taxco's customers is equal to or greater than $150.
Alternative hypothesis (Ha): The average tax refund for Taxco's customers is less than $150.
Next, we need to calculate the test statistic and determine the critical value for the test. Since we have a small sample size (n=6), we will use the t-test statistic.
To calculate the t-test statistic, we need the sample mean, the population mean, the sample standard deviation, and the sample size.
Given data:
Sample mean (x̄) = (144+38+130+160+81+135)/6 = 108.33
Population mean (μ) = $150
Sample standard deviation (s) = sqrt(((144-108.33)^2 + (38-108.33)^2 + (130-108.33)^2 + (160-108.33)^2 + (81-108.33)^2 + (135-108.33)^2) / (6-1)) = 48.33
Sample size (n) = 6
Now, we can calculate the t-test statistic using the formula:
t = (x̄ - μ) / (s / sqrt(n))
t = (108.33 - 150) / (48.33 / sqrt(6)) ≈ -1.23
The critical value can be obtained from the t-distribution table or using statistical software. At a 5% level of significance and degrees of freedom = n-1 = 5, the critical t-value is approximately -2.571.
Comparing the t-test statistic (-1.23) with the critical value (-2.571), we find that the t-test statistic is greater than the critical value. This means that the t-test does not fall in the critical region.
Since the t-test does not fall in the critical region, we fail to reject the null hypothesis. Therefore, there is not enough evidence to support the specialist's belief that Taxco has a lower average tax refund for its customers.