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 ![s = \sqrt{\frac{p(1-p)}{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D)
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
![\pi_1 = 0.49, s_{\pi_1} = \sqrt{\frac{0.49*0.51}{600}} = 0.0204](https://tex.z-dn.net/?f=%5Cpi_1%20%3D%200.49%2C%20s_%7B%5Cpi_1%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B0.49%2A0.51%7D%7B600%7D%7D%20%3D%200.0204)
Males:
41% from a sample of 500. So
![\pi_2 = 0.41, s_{\pi_2} = \sqrt{\frac{0.41*0.59}{500}} = 0.022](https://tex.z-dn.net/?f=%5Cpi_2%20%3D%200.41%2C%20s_%7B%5Cpi_2%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B0.41%2A0.59%7D%7B500%7D%7D%20%3D%200.022)
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
![H_0: \pi_1 - \pi_2 = 0](https://tex.z-dn.net/?f=H_0%3A%20%5Cpi_1%20-%20%5Cpi_2%20%3D%200)
At the alternate hypothesis, we test if there is a difference, that is, the subtraction of the proportions is different of 0. So
![H_1: \pi_1 - \pi_2 \neq 0](https://tex.z-dn.net/?f=H_1%3A%20%5Cpi_1%20-%20%5Cpi_2%20%5Cneq%200)
The test statistic is:
![z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
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 ![\mu = 0](https://tex.z-dn.net/?f=%5Cmu%20%3D%200)
From the two samples:
![X = \pi_1 - \pi_2 = 0.49 - 0.41 = 0.08](https://tex.z-dn.net/?f=X%20%3D%20%5Cpi_1%20-%20%5Cpi_2%20%3D%200.49%20-%200.41%20%3D%200.08)
![s = \sqrt{s_{\pi_1}^2 + s_{\pi_2}^2} = \sqrt{0.0204^2 + 0.022^2} = 0.03](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bs_%7B%5Cpi_1%7D%5E2%20%2B%20s_%7B%5Cpi_2%7D%5E2%7D%20%3D%20%5Csqrt%7B0.0204%5E2%20%2B%200.022%5E2%7D%20%3D%200.03)
Value of the test statistic:
![z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![z = \frac{0.08 - 0}{0.03}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B0.08%20-%200%7D%7B0.03%7D)
![z = 2.67](https://tex.z-dn.net/?f=z%20%3D%202.67)
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.