Question

"The Craven," by Mark Salangsang

Once upon a morning dreary

In stats class I was weak and weary.

Pondering over last night’s homework

Whose answers were now on the board

This I did and nothing more.

While I nodded nearly napping

Suddenly, there came a tapping.

As someone gently rapping,

Rapping my head as I snore.

Quoth the teacher, “Sleep no more.”

“In every class you fall asleep,”

The teacher said, his voice was deep.

“So a tally I’ve begun to keep

Of every class you nap and snore.

The percentage being forty-four.”

“My dear teacher I must confess,

While sleeping is what I do best.

The percentage, I think, must be less,

A percentage less than forty-four.”

This I said and nothing more.

“We’ll see,” he said and walked away,

And fifty classes from that day

He counted till the month of May

The classes in which I napped and snored.

The number he found was twenty-four.

At a significance level of 0.05,

Please tell me am I still alive?

Or did my grade just take a dive

Plunging down beneath the floor?

Upon thee I hereby implore.

Short answer

Confidence Interval: (0.3415,0.6185).

Step-by-step solution

Find H0 and Ha: We want to test if Carven takes less than 44% of naps and snores in every class, on average.

$H_0:p=0.44;H_a:p<0.44$

Determine the distribution needed: In words, CLEARLY state what your random variable P′ represents.Let P’ = the proportion of classes in which the student falls asleep.

Normal:$N\left(0.44,\sqrt{\frac{(0.44)(1-0.44)}{50}}\right)$

Test Statistic:$z=0.57$

Calculate the p-value using the normal distribution for proportions: 

$p-value=0.7156$

In one to two complete sentences, explain what the p-value means for this problem. If the null hypothesis is true (the proportion is 0.44), then there is a 0.7156 probability that the sample (estimated) proportion is 0.44 or more.

Compare α and the p-value:Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.

alpha	decision	reason for decision
0.05	Do not reject the null hypothesis.	$p-value>0.05$

Conclusion: There is insufficient evidence to support the claim that the student falls asleep in less than 44% of classes.

Confidence interval

Confidence Interval: (0.3415,0.6185). The “plus-4s” confidence interval is (0.3482, 0.6148).