Respuesta :
Answer:
Step-by-step explanation:
This is a test of 2 population proportions. Let 1 and 2 be the subscript for the women and men. The population proportion of women and men who favor spending more federal tax dollars on the arts would be p1 and p2 respectively.
P1 - P2 = difference in the proportion of women and men who favor spending more federal tax dollars on the arts.
The null hypothesis is
H0 : p1 = p2
p1 - p2 = 0
The alternative hypothesis is
Ha : p1 ≠ p2
p1 - p2 ≠ 0
it is a two tailed test
Sample proportion = x/n
Where
x represents number of success(number of complaints)
n represents number of samples
For women
x1 = 57
n1 = 209
P1 = 57/209 = 0.27
For men,
x2 = 59
n2 = 194
P2 = 59/194 = 0.3
The pooled proportion, pc is
pc = (x1 + x2)/(n1 + n2)
pc = (57 + 59)/(209 + 194) = 0.29
1 - pc = 1 - 0.29 = 0.71
z = (P1 - P2)/√pc(1 - pc)(1/n1 + 1/n2)
z = (0.27 - 0.3)/√(0.29)(0.71)(1/209 + 1/194) = - 0.03/0.045
z = - 0.67
Since it is a two tailed test, the curve is symmetrical. We will look at the area in both tails. Since it is showing in one tail only, we would double the area
From the normal distribution table, the area below the test z score in the left tail 0.25
We would double this area to include the area in the right tail of z = 0.67 Thus
p = 0.25 × 2 = 0.5
By using the p value,
Since 0.1 < 0.5, we would accept the null hypothesis.
By using the critical region method,
The calculated test statistic is 0.67 for the right tail and - 0.67 for the left tail
Since α = 0.1, the critical value is determined from the normal distribution table.
For the left, α/2 = 0.1/2 = 0.05
The z score for an area to the left of 0.05 is - 1.645
For the right, α/2 = 1 - 0.05 = 0.95
The z score for an area to the right of 0.95 is 1.645
In order to reject the null hypothesis, the test statistic must be smaller than - 1.645 or greater than 1.645
Since - 0.67 > - 1.645 and 0.67 < 1.645, we would fail to reject the null hypothesis.
Answer:
The null hypothesis failed to be rejected.
There is not enough evidence to support the claim that the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts differs significantly.
Test statistic z = 0.70
P-value = 0.4871
Step-by-step explanation:
This is a hypothesis test for the difference between proportions.
The claim is that the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts differs significantly.
Then, the null and alternative hypothesis are:
[tex]H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\neq 0[/tex]
The significance level is 0.10.
The sample 1, of size n1=209 has a proportion of p1=0.2727.
[tex]p_1=X_1/n_1=57/209=0.2727[/tex]
The sample 2, of size n2=194 has a proportion of p2=0.3041.
[tex]p_2=X_2/n_2=59/194=0.3041[/tex]
The difference between proportions is (p1-p2)=-0.0314.
[tex]p_d=p_1-p_2=0.2727-0.3041=-0.0314[/tex]
The pooled proportion, needed to calculate the standard error, is:
[tex]p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{57+59}{209+194}=\dfrac{116}{403}=0.2878[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.2878*0.7122}{209}+\dfrac{0.2878*0.7122}{194}}\\\\\\s_{p1-p2}=\sqrt{0.00098+0.00106}=\sqrt{0.00204}=0.0451[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{-0.0314-0}{0.0451}=\dfrac{-0.0314}{0.0451}=-0.696[/tex]
This test is a two-tailed test, so the P-value for this test is calculated as (using a z-table):
[tex]P-value=2\cdot P(t<-0.696)=0.4871[/tex]
As the P-value (0.4871) is bigger than the significance level (0.1), the effect is not significant.
The null hypothesis failed to be rejected.
There is not enough evidence to support the claim that the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts differs significantly.
