Question

3. Suppose someone claims that the average GPA in a University is at least 2.85.
But in a class with 23 students, we find the average GPA in that class is 2.78,
with standard deviation 0.35. Can we reject the claim at significance level 0.10
?

4. Is the random machine really random? Suppose to check the random
machine, we let the machine to produce 10,000 digit numbers (0, 1, ..., 9). We
find the number of 1’s is 1080. Can we say with α=0.05, that the machine is
not really random?

5. Suppose someone claims that the average GPA for all students in this
university is at least 3.00. To test this hypothesis, we set H_0: μ=3.0, H_1:
μ<3.0. Suppose a professor asks you to check a random sample of 36 students,
and tells you that if the sample mean is less than 2.95, then, reject the claim.
For this decision making rule, find α, the probability for making Type I error
(assume the standard deviation of GPA for all students is 0.4).

Answer 1

3. Suppose someone claims that the average GPA in a University is at least 2.85.
But in a class with 23 students, we find the average GPA in that class is 2.78,
with standard deviation 0.35. Can we reject the claim at significance level 0.10
?

Sample size = n = 23

Sample mean = = 2.78

Standard deviation = s = 0.35

Claim: The average GPA in a University is at least 2.85.

The null and alternative hypothesis is

Level of significance = 0.10

Here population standard deviation is unknown so we have to use t-test statistic.
Test statistic is

Degrees of freedom = n - 1 = 23 - 1 = 22

Critical value = 1.321 ( Using t table)

Test statistic | t | < critical vaue we fail to reject null hypothesis.

Conclusion:

The average GPA in a University is at least 2.85.

----------------------------------------------------------------------------------------------------

For the remaining questions please repost!