Question

Math

Posted 4 months ago

```
According to a large poll in a previous year, about $80 \%$ of homes in a certain county had access to high-speed internet. Market researchers wondered if that proportion had changed, so they took a random sample of 64 homes from that county and found that 48 of them had access to highspeed internet.
They want to use this sample data to test $H_{0}: p=0.8$ versus $H_{\mathrm{a}}: p \neq 0.8$, where $p$ is the proportion of homes in this county with high-speed internet access.
Assuming that the conditions for inference have been met, calculate the test statistic for their significance test.
You may round to two decimal places.
$z=$
```

Sign in to unlock the answer

Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Calculate the sample proportion $\hat{p}$ by dividing the number of homes with high-speed internet by the total number of homes in the sample

$\hat{p} = \frac{48}{64}$

step 2

Compute the standard error (SE) using the null hypothesis proportion $p_0$ and the sample size $n$

$SE = \sqrt{\frac{p_0(1 - p_0)}{n}}$ where $p_0 = 0.8$ and $n = 64$.

step 3

Calculate the z-test statistic using the sample proportion $\hat{p}$, the null hypothesis proportion $p_0$, and the standard error SE

$z = \frac{\hat{p} - p_0}{SE}$

step 4

Substitute the values into the z-test statistic formula and calculate

$z = \frac{\frac{48}{64} - 0.8}{\sqrt{\frac{0.8(1 - 0.8)}{64}}}$

step 5

Simplify the calculations to find the z-test statistic

$z = \frac{0.75 - 0.8}{\sqrt{\frac{0.8 \cdot 0.2}{64}}}$

$z = \frac{-0.05}{\sqrt{\frac{0.16}{64}}}$

$z = \frac{-0.05}{0.05}$

$z = -1$

Answer

The test statistic for their significance test is $z = -1$.

Key Concept

Hypothesis Testing and Z-Test Statistic

Explanation

The z-test statistic is used in hypothesis testing to determine how many standard errors a sample proportion is from the hypothesized population proportion under the null hypothesis. It is calculated by subtracting the null hypothesis proportion from the sample proportion, then dividing by the standard error of the sample proportion.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question