Question 1192947: 16. Ford Motor Company claims that its new 8-cylinder sedan, the XSG, will average 24 miles per gallon.
Recently, on the blog Ford#1, the show’s host collected sample gas mileage information randomly from 29
XSG owners who follow the blog. The sample revealed a mean of 21.5 mpg with standard deviation 1.9 mpg. Use correct symbols. AND THEN Compute the sample mean and standard deviation.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! population mean is assumed to be 24.
sample mean is 21.5
sample standard deviation is 1.9.
sample size is 29.
standard error is 1.9 / sqrt(29) = .352821 rounded to 6 decimal places.
z = (x - m) / s
z is the z-score
x is the sample mean
m is the population mean
s is the standard error.
the formula becomes z = (21.5 - 24) / .352821 = -7.08574603.
the area to the left of that z-score is equal to 0.
what this says is that the probability you would get a z-score less than -7.08..... is so small as it is effectively equal to 0.
here's a reference on symbols.
https://www.dummies.com/article/academics-the-arts/math/statistics/seeing-what-statistical-symbols-stand-for-142633/
note that, because you are using the sample standard deviation rather than the population standard deviation, you would probably be using the t-stoce rather than the z-score.
the t-score itself is calculated in the same way as the z-score.
the difference is in the area under the normal distribution curve to the left or right of the t-score as opposed to the area to the left or right of the z-score.
t-scores use degrees of freedom whereas z-score don't.
the degrees of freedom are usually 1 less than the sample size.
if the degrees of freedom are small, there is a larger difference between the area to the left or right of a t-score compared to the area to the left or right of a z-score.
some examples:
z-score = -1
area to the left = .158655
t-score with 5 degrees of freedom = -1
area to the left = .181609
t-score with 50 degrees of freedom = -1
area to the left = .161063
t-score with 1000 degrees of freedom = -1
area to the left = .158776
as the degrees of freedom get greater, the area to the left of the t-score approaches the area to the left of the z-score.
the difference is most pronounced when the degrees of freedom are small.
here's a reference that shows you the difference between a t distribution and a z distribution.
https://www.jmp.com/en_us/statistics-knowledge-portal/t-test/t-distribution.html#:~:text=What's%20the%20key%20difference%20between,on%20the%20sample%20standard%20deviation.
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