Melissa Hart, a quantitative analyst with the Kansas City Federal Reserve, has been involved with accumulating data for an essay regarding economic conditions in her area. Specifically, Melissa has been gathering data related to the levels of consumer debt in the state of Missouri. The following is a description of the data:
Sample mean: $3,451.00 -
Standard deviation of the sample: $819
Number of observations: 441 -
Assume that Ms. Hart formulates a hypothesis test in which the null hypothesis specifies that the population mean is equal to $4,600. Additionally, assume that the population standard deviation is unknown. Given this information, what is the standard error of the estimate? Further, what is the test statistic? Choose the best answer.
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If the population standard deviation is unknown, as in this example, the standard error of the estimate is found by using the following equation:
{Standard error = s / square root of n} where s = the sample standard deviation and n = the number of observations in the sample.
In this example, all of the necessary information has been provided, and the determination of the standard error of the estimate is found as:
{Standard error = [$819 / 21] = $39}
Now that the standard error of the estimate has been calculated, the test statistic can be found by using the following equation: {Test statistic = [sample statistic - value of the population parameter under the null hypothesis] / standard error of the sample statistic]. Again, all of the necessary information has been provided, and the calculation of the test statistic is found as follows:
{Test statistic = [$3,451 - $4,600] / $39 = (29.46)}
This is a rather large test statistic, and will likely result in the rejection of the null hypothesis unless a very low level of confidence is employed.