Answer:
a) The p-value of the test is 0.0076 < 0.05, which means that there is evidence of a difference between males and females in the proportion who said they prefer window tinting as a luxury upgrade at the 0.05.
b) The null hypothesis is and the alternate hypothesis is
.
Step-by-step explanation:
Before testing the hypothesis, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
Females:
49% from a sample of 600. So
Males:
41% from a sample of 500. So
Test if there is a difference between males and females in the proportion who said they prefer window tinting as a luxury upgrade.
From here, question b can already be answered.
At the null hypothesis we test if there is no difference, that is, the subtraction of the proportions is 0. So
At the alternate hypothesis, we test if there is a difference, that is, the subtraction of the proportions is different of 0. So
The test statistic is:
In which X is the sample mean, is the value tested at the null hypothesis, and s is the standard error.
0 is tested at the null hypothesis:
This means that
From the two samples:
Value of the test statistic:
Question a:
P-value of the test and decision:
The p-value of the test is the probability that the sample proportion differs from 0 by at least 0.08, which is P(|Z| > 2.670, which is 2 multiplied by the p-value of Z = -2.67.
Looking at the z-table, Z = -2.67 has a p-value of 0.0038.
2*0.0038 = 0.0076
The p-value of the test is 0.0076 < 0.05, which means that there is evidence of a difference between males and females in the proportion who said they prefer window tinting as a luxury upgrade at the 0.05.