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Answer 1

Answer:

The null hypothesis is $H_0: p \geq 0.2$

The alternative hypothesis is $H_1: p < 0.2$

The p-value of the test is of 0.0154 > 0.01, which means that at the 0.01 significance level, there is not significant evidence that the return rate is less than 20%.

Step-by-step explanation:

Test the claim that the return rate is less than 20%.

At the null hypothesis, we test if the return rate is of at least 20%, that is:

$H_0: p \geq 0.2$

At the alternative hypothesis, we test if the return rate is less than 20%, that is:

$H_1: p < 0.2$

Normal distribution as an approximation to the binomial distribution to find the Z-score.

Binomial probability distribution

Probability of exactly x successes on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$, $\sigma = \sqrt{V(X)}$.

0.2 is tested at the null hypothesis. Sample of 6957.

This means that $p = 0.2, n = 6957$. So

$\mu = E(X) = np = 6957*0.2 = 1391.4$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{6957*0.2*0.8} = 33.36$

1319 surveys returned

This means that, due to continuity correction, $X = 1319 + 0.5 = 1319.5$. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1319.5 - 1391.4}{33.36}$

$Z = -2.16$

P-value of the test:

The p-value of the test is the probability of 1319 or less people returning the survey, which is the p-value of Z = -2.16.

Looking at the z-table, Z = -2.16 has a p-value of 0.0154.

The p-value of the test is of 0.0154 > 0.01, which means that at the 0.01 significance level, there is not significant evidence that the return rate is less than 20%.