# (Test on a Mean, Sigma Unknown): A random sample of 12 first-year university students enrolled in...

(Test on a Mean, Sigma Unknown): A random sample of 12 first-year university students enrolled in MATH 1051H was drawn. At the course's completion, each student was asked how many hours he or she spent doing homework in statistics. The data are listed below.
(28, 36, 38, 37, 32,36), (31, 31, 40, 40, 41, 36)
You estimate that the population standard deviation is 7. The instructor has recommended that students devote 3 hours per week for the duration of the 12-week semester, for a total of 36 hours. Test to determine whether there is evidence at the 0.01 significance level that the average student spent less than the recommended amount of time. Write a complete hypothesis test solution?

$Let~\mu=true~average~amount~of~time~spent~for~doing~homework~in~statistics.\\ Null~hypothesis,~H_0:\mu=36~vs.~Alternative~hypothesis,~H_1:\mu<36.$

R code:
x=c(28, 36, 38, 37, 32,36,31, 31, 40, 40, 41, 36)
popsd=7
mu=36 #hypothesized value
z_score <- round((mean(x)-mu)/(popsd/sqrt(length(x))),3)
p_value <- round(pnorm(z_score),3)
z_score #test statistic
p_value #p-value

Output:

> z_score #test statistic
[1] -0.247
> p_value #p-value
[1] 0.402

Since p-value>0.01, so we reject H0 at 1% level of significance and conclude that there is insufficient evidence to conclude that the average student spent less than the recommended amount of time.

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