Answer:
The null hypothesis is
The alternative hypothesis is
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:
At the alternative hypothesis, we test if the return rate is less than 20%, that is:
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:
The standard deviation of the binomial distribution is:
Normal probability distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score of a measure X is given by:
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 ,
.
0.2 is tested at the null hypothesis. Sample of 6957.
This means that . So
1319 surveys returned
This means that, due to continuity correction, . So
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%.