SOLUTION: It takes runners on average 289 minutes to run the marathon. The standard deviation of the times to run the marathon is known to be 26 minutes. Drink ZXZ manufacturer claims that

Question 1199108: It takes runners on average 289 minutes to run the marathon. The standard deviation of the times to run the marathon is known to be 26 minutes.
Drink ZXZ manufacturer claims that drinking ZXZ helps runners to run the
marathon faster. To test that claim, we took 30 runners, and gave them to drink
ZXZ before they ran the marathon. In our sample, the average time to run the
marathon was 281 minutes.
Part a) What is H0 and what is H1 in this test?
Part b) Do we have enough evidence to reject H0 and accept H1?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the population mean is assumed to be 289 minutes and the population standard deviation is assumed to be 26 minutes.

the sample size is 30 runners.
the sample mean is 281 minutes.

H0 says that the average mean is what the assumed population mean is (289)
H1 says that the average mean is not 289.

since you are looking at the mean of the sample, you use the standard error rather than the standard deviation.
standard error = standard deviation / square root of sample size = 26 / sqrt(30) = 4.7469 rounded to 4 decimal places.

use the z-score formula to find the z-score.
the z-score formula is z = (x - m) / s
in this case, .....
z is the z-score
x is the mean of the sample
m is the assumed population mean.
s is the standard error.

you get z = (281 - 289) / 4.7469 = -1.6893 rounded to 4 decimal places.

the area to the left of that z-score is equal to .0456 rounded to 4 decimal places.

since your test was for not equal, you would have a two tailed confidence interval.
at 95% confidence interval, the tail on each end would be 2.5% = .025.
since the test tail on the left was .0456, this is greater thaan the critical tail at .025.
consequently you would assume that you did not have enough information to conclude that the population mean was not equal to 289.
the difference would be assumed to be caused by random variation in the sample mean.

here's what the test results would look like on the z-score calculator at https://davidmlane.com/hyperstat/z_table.html

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