A telephone company's records indicate that private customers pay on average $17.10 per month for long-distance telephone calls. A random sample of 10 customers' bills during a given month produced a sample mean of $22.10 expended for long-distance calls and a sample variance of 45. A 5% significance test is to be performed to determine if the mean level of billing for long distance calls per month is in excess of $17.10. The calculated value of the test statistic and the critical value respectively are:
.05,.07
To determine if the mean level of billing for long-distance calls per month is in excess of $17.10, we can perform a hypothesis test.
Let's define the null and alternative hypotheses:
Null hypothesis (H0): The mean level of billing for long-distance calls per month is $17.10 or less.
Alternative hypothesis (Ha): The mean level of billing for long-distance calls per month is in excess of $17.10.
To perform the test, we need to calculate the test statistic and compare it with the critical value.
The test statistic for this scenario is the t-test statistic, which is given by the formula:
t = (sample mean - population mean) / (sample standard deviation / √sample size)
Given that the sample mean is $22.10, the population mean is $17.10, the sample variance is 45, and the sample size is 10, we can calculate the sample standard deviation using the sample variance:
sample standard deviation = √sample variance = √45 ≈ 6.71
Plugging these values into the formula, we get:
t = (22.10 - 17.10) / (6.71 / √10) ≈ 4.38
Now, to determine the critical value for a 5% significance level and 9 degrees of freedom (sample size minus 1), we consult the t-distribution table or use statistical software. For a one-tailed test (greater than), the critical value is approximately 1.833.
Comparing the calculated test statistic (4.38) to the critical value (1.833), we can conclude that the calculated value of the test statistic is greater than the critical value.
In summary, the calculated value of the test statistic is 4.38, and the critical value (for a 5% significance level and 9 degrees of freedom) is 1.833.