Chapter 10: Q 22. (page 641)

Are teenagers going deaf? In a study of $3000$ randomly selected teenagers in $1990, 450$ showed some hearing loss. In a similar study of 1800 teenagers reported in $2010, 351$ showed some hearing loss.
a. Do these data give convincing evidence that the proportion of all teens with hearing
loss has increased at the $α = 0.01$ significance level?
b. Interpret the $P$ -value from part (a) in the context of this study.

Short Answer

Expert verified

a. There is convincing evidence that the proportion of all teens with hearing loss has increased over the years.

b. Probability is approximately equal to zero.

Step by step solution

Given Information

It is given that $x_{1} = 450$

$x_{2} = 351$

$n_{1} = 3000$

$n_{2} = 1800$

$α = 0.01$

Claim: Population of all teens with hearing loss is higher in $2010$

To explain if data give convincing evidence that the proportion of all teens with hearing loss has increased at the α=0.01 significance level or not.

The hypothesis is:

Null: $H_{0} : p_{1} = p_{2}$

Alternative: $H_{a} : p_{1} < p_{2}$

$p_{1}$ is proportion of teens in $1990$ with hearing loss and $p_{2}$ is \text { proportion of teens in $2010$ with hearing loss.

Conditions are:

Random: Teenagers are independent random samples.

Independent: $3000 < 10 % in 1990$ and $1800 < 10 % in 2010$

Normal: Success are $450, 351$ and failures are $3000 - 450 = 2550$ and $1800 - 351 = 1449$ . All are greater than ten.

All condition are satisfied. Hypothesis test can be performed.

Sample Proportion: ${\hat{p}}_{1} = \frac{x_{1}}{n_{1}} = \frac{450}{3000} = 0.15$

${\hat{p}}_{2} = \frac{x_{2}}{n_{2}} = \frac{351}{1800} = 0.195$

and ${\hat{p}}_{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}} = \frac{450 + 351}{3000 + 1800} = \frac{800}{4800} = 0.1667$

Test statistics:

$z = \frac{{\hat{p}}_{1} - {\hat{p}}_{2} - (p_{1} - p_{2})}{\sqrt{{\hat{p}}_{p} (1 - {\hat{p}}_{p})} \sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}} = \frac{0.15 - 0.195 - 0}{\sqrt{0.1667 (1 - 0.1667)} \sqrt{\frac{1}{3000} + \frac{1}{1800}}} \approx - 4.05$

Probability: $P = P (Z < - 4.05) \approx 0$

Also, $P < 0.01 \Rightarrow Reject H_{0}$

So, there is convincing evidence that the proportion of all teens with hearing loss has increased over the years.

Step 3: P value

As $P \approx 0$

There is a very small chance of obtaining similar test results or more extreme when the proportion of all teens with hearing loss is greater in $2010$ compared to $1990$ .