Question 1188602
the population mean is assumed to be 30,000 (national average).
the population standard deviation is not known, therefore the sample standard deviation is used instead and the t-score is used instead of the z-score.


here's a reference on choosing between z-score and t-score.
<a href = "https://math.stackexchange.com/questions/1817980/how-to-know-when-to-use-t-value-or-z-value" target = "_blank">https://math.stackexchange.com/questions/1817980/how-to-know-when-to-use-t-value-or-z-value</a>


the statistics for the sample given was determined by use of the online descriptive statistics calculator at <a href = "https://www.calculatorsoup.com/calculators/statistics/descriptivestatistics.php" target = "_blank">https://www.calculatorsoup.com/calculators/statistics/descriptivestatistics.php</a>


from this calculator, it was determined that:


the sample mean was 33949.5238.
the sample standard deviation was 11866.174.
the sample size was 21.


10% level of significance equals 90% confidence interval.
two tail 90% confidence interval has 5% tail on each end.
critical t-score with 20 degrees of freedom (sample size minus 1) = plus or minus 1.742718218.


this was determined through the use of the ti-84 plus scientific calculator, but could also have been determined through the online calculator found at <a href = "https://www.omnicalculator.com/statistics/critical-value" target = "_blank">https://www.omnicalculator.com/statistics/critical-value</a>


at critical t-score of plus or minus 1.742718218, use the t-score formula to find the raw score.


that formula is t = (x - m) / s
t is the t-score.
x is the sample mean.
m is the assumed population mean.
s is the standard error.


the standard error is the standard deviation of the distribution of sample means.
the formula for that is:
s = standard deviation / square root of sample size.
that becomes:
s = 11866.174 / sqrt(21) = 2589.411455.


the test t-score was determined to be:


t = (33949.5238 - 30000) / 2589.41455 = 1.525257437.


since this was less than the critical t-score of 1.742718218, the results of the test were not significant, indicating there was not enough evidence to say that the mean scores for the dealership were different than the mean scores of the population.


here's a display of the descriptive statistics for the sample.


<img src = "http://theo.x10hosting.com/2021/121001.jpg" >


here's a display of the results of using the online critical t-value calculator.


<img src = "http://theo.x10hosting.com/2021/121002.jpg" >