Question

"Phillip’s Wish," by Suzanne Osorio

My nephew likes to play

Chasing the girls makes his day.

He asked his mother

If it is okay

To get his ear pierced.

She said, “No way!”

To poke a hole through your ear,

Is not what I want for you, dear.

He argued his point quite well,

Says even my macho pal, Mel,

Has gotten this done.

It’s all just for fun.

C’mon please, mom, please, what the hell.

Again Phillip complained to his mother,

Saying half his friends (including their brothers)

Are piercing their ears

And they have no fears

He wants to be like the others.

She said, “I think it’s much less.

We must do a hypothesis test.

And if you are right,

I won’t put up a fight.

But, if not, then my case will rest.”

We proceeded to call fifty guys

To see whose prediction would fly.

Nineteen of the fifty

Said piercing was nifty

And earrings they’d occasionally buy.

Then there’s the other thirty-one,

Who said they’d never have this done.

So now this poem’s finished.

Will his hopes be diminished,

Or will my nephew have his fun?

Short answer

Confidence Interval: (0.245, 0.515)

Step-by-step solution

Step 1: Find H0 and Ha: We want to test if guys take less than 50% of the piercing ears, on average.

$H_0:p=0.50;H_a:p<0.50$

Determine the distribution needed: In words, CLEARLY state what your random variable P′ represents.Let P’ = the proportion of friends that has a pierced ear

Normal:$N\left(0.5,\sqrt{\frac{(0.5)(1-0.5)}{31}}\right)$

Test Statistic:role="math" localid="1649422725054" $z=–1.70$

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

$p-value=0.0448$

In one to two complete sentences, explain what the p-value means for this problem. If the null hypothesis is true (the proportion is 0.50), then there is a 0.04408 probability that the sample (estimated) proportion is 0.38 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	Reject the null hypothesis	$p-value<0.05$

Conclusion: There is sufficient evidence to support the claim that less than 50% of his friends have pierced ears.

Confidence Interval

Confidence Interval: (0.245, 0.515): The “plus-4s” confidence interval is (0.259, 0.519).