Respuesta :
Answer:
The p-value of the test is 0.242 > 0.05, which means that this information does not indicate a difference between the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts.
Step-by-step explanation:
Before solving this question, 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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
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 [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
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.
Women:
51 out of 222, so:
[tex]p_1 = \frac{51}{222} = 0.2297[/tex]
[tex]s_1 = \sqrt{\frac{0.2297*0.7703}{222}} = 0.0282[/tex]
Men:
49 out of 174, so:
[tex]p_2 = \frac{49}{174} = 0.2816[/tex]
[tex]s_2 = \sqrt{\frac{0.2816*0.7184}{174}} = 0.0341[/tex]
Does this information indicate a difference (either way) between the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts?
Either way, so a two tailed test to see if the difference of proportions is different of 0.
At the null hypothesis, we test if it is not different of 0, so:
[tex]H_0: p_1 - p_2 = 0[/tex]
At the alternative hypothesis, we test if it is different of 0, so:
[tex]H_1: p_1 - p_2 \neq 0[/tex]
The test statistic is:
[tex]z = \frac{X - \mu}{s}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, and s is the standard error.
0 is tested at the null hypothesis:
This means that [tex]\mu = 0[/tex]
From the samples:
[tex]X = p_1 - p_2 = 0.2297 - 0.2816 = -0.0519[/tex]
[tex]s = \sqrt{s_1^2+s_2^2} = \sqrt{0.0282^2+0.0341^2} = 0.0442[/tex]
Value of the test statistic:
[tex]z = \frac{X - \mu}{s}[/tex]
[tex]z = \frac{-0.0519 - 0}{0.0442}[/tex]
[tex]z = -1.17[/tex]
P-value of the test and decision:
The p-value of the test is the probability of the differences being of at least 0.0519, either way, which is P(|z| > 1.17), that is, 2 multiplied by the p-value of z = -1.17.
Looking at the z-table, z = -1.17 has a p-value of 0.121.
0.121*2 = 0.242
The p-value of the test is 0.242 > 0.05, which means that this information does not indicate a difference between the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts.
