A recent article in a computer magazine suggested that the mean time to fully learn a new software program is 40 hours. A sample of 100 first time users of a new statistics program revealed the mean time to learn it was 39 hours with the standard deviation of 5 hours. At the 0.05 significance level, can we conclude that users learn the package in less than a mean of 40 hours?

a. State the null and alternate hypotheses.
Ho:
H1:
b. State the decision rule.
c. Compute the value of the test statistic.
d. Compute the p-value.
e. What is your decision regarding the null hypothesis? Interpret the result.

Answered in another post.

a. The null hypothesis (Ho) is that users learn the package in a mean time of 40 hours or more.

The alternate hypothesis (H1) is that users learn the package in less than a mean of 40 hours.

b. The decision rule is to reject the null hypothesis if the p-value is less than the significance level (0.05).

c. To compute the value of the test statistic, we can use the formula for a one-sample t-test:
t = (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size))
t = (39 - 40) / (5 / sqrt(100))
t = -1 / (5 / 10)
t = -2

d. To compute the p-value, we need to find the probability of observing a test statistic as extreme as -2 (or more extreme) assuming that the null hypothesis is true. We can do this by looking up the t-distribution table or using statistical software:
The p-value for t = -2 with degrees of freedom (df) = n - 1 = 100 - 1 = 99 is approximately 0.0287.

e. Comparing the p-value (0.0287) to the significance level (0.05), we see that the p-value is less than the significance level. Therefore, we reject the null hypothesis.
We can conclude that there is sufficient evidence to suggest that users learn the package in less than a mean of 40 hours.