Puppies need to sleep for at least 7 hours a night. We are trying to find whether most puppies meet or exceed this amount of sleep. A random sample of 25 puppies is taken and we observe a mean of 6.501 hours with a standard deviation of 1 hour.

1. Write a 90% confidence interval for the mean hours of sleep that of puppies.

2. Assuming the same standard deviation of 1 hour, how many students should be observed to decrease the margin of error by 50%? (You don’t know the degree of freedom as you don’t know the sample size. You should first find an approximate sample size n using the Z-model (normal mode) then once you have the n you can find the corresponding t* to be used in ME. Now use that t* to recalculate the sample size n.

3. Is there evidence that puppies sleep less than 7 hours? What are the appropriate hypotheses? Perform the t-test and find the P-value. Use a 5% significance level.

4. Why do we reject the null hypothesis if p-value is less than the significance level? Give the intuition of the p-value and explain the meaning.

Please prove why this is correct and give a detailed explanation.