Respuesta :
Answer:
There is not enough evidence to reject the null hypothesis.
Step-by-step explanation:
(a)
The hypothesis can be defined as follows:
H₀: p₁ - p₂ ≤ 0 vs. Hₐ: p₁ - p₂ > 0.
(b)
The test statistic is defined as follows:
[tex]z=\frac{\hat p_{1}-\hat p_{2}}{\sqrt{\hat P(1-\hat P)[\frac{1}{n_{1}}+\frac{1}{n_{2}}]}}[/tex]
The information provided is:
n₁ = 244
n₂ = 311
x₁ = 122
x₂ = 137
Compute the sample proportions and total proportions as follows:
[tex]\hat p_{1}=\frac{x_{1}}{n_{1}}=\frac{122}{244}=0.50\\\\\hat p_{2}=\frac{x_{2}}{n_{2}}=\frac{137}{311}=0.44\\\\\hat P=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{122+137}{244+311}=0.47[/tex]
Compute the test statistic value as follows:
[tex]z=\frac{\hat p_{1}-\hat p_{2}}{\sqrt{\hat P(1-\hat P)[\frac{1}{n_{1}}+\frac{1}{n_{2}}]}}[/tex]
[tex]=\frac{0.50-0.44}{\sqrt{0.47(1-0.47)[\frac{1}{244}+\frac{1}{311}]}}\\\\=1.41[/tex]
The test statistic value is 1.41.
The decision rule is:
The null hypothesis will be rejected if the p-value of the test is less than the significance level α = 0.05.
Compute the p-value as follows:
[tex]p-value=P(Z>1.41)\\=1-P(Z<1.41)\\=1-0.92073\\=0.07927\\\approx 0.08[/tex]
*Use a z-table.
The p-value of the test is 0.08.
p-value = 0.08 > α = 0.05
The null hypothesis will not be rejected at 5% significance level.
Thus, there is not enough evidence to reject the null hypothesis.
