Answer:
Step-by-step explanation:
Hello!
You need to test at 5% that the proportion of women that purchased online past month is greater than the proportion of men that purchased online past month.
There are two study variables:
X₁: Number of men that purchased online past month, out of 225
n₁= 225
p'₁= 0.40
X₂: Number of women that purchased online past month, out of 276
n₂= 276
p'₂= 0.50
Since both variables met the Binomial criteria:
1. The number of observations of the trial is fixed.
2. The experiment has only two possible outcomes, "success" and "failure"
3. Each observation in the trial is independent, this means that none of the trials will affect the probability of the next trial.
4. The probability of success in the same from one trial to another.
We can say that X₁~ Bi(n₁;ρ₁) and X₂~ Bi(n₂;ρ₂)
Since n₁≥30; n₁*ρ₁≥5; n₁*(1-ρ₁)≥5 and n₂≥30; n₂*ρ₂≥5; n₂(1-ρ₂)≥5 it is valid to use the Central Limit Theorem to approximate the distribution of both sample proportions p'₁ and p'₂ to normal and use the approximation of the standard normal with pooled sample proportion to analyze if the claim is true or not.
The statistic hypotheses are:
H₀: ρ₁ ≥ ρ₂
H₁: ρ₁ < ρ₂
α: 0.05
This test is one-tailed left, the critical value is:
![Z_{\alpha }= Z_{0.05}= -1.648](https://tex.z-dn.net/?f=Z_%7B%5Calpha%20%7D%3D%20Z_%7B0.05%7D%3D%20-1.648)
You will reject the null hypothesis to values of
≤ -1.648
![Z_{H_0}= \frac{(p'_1-p'_2)-(p_1-p_2)}{\sqrt{p'(1-p')*(\frac{1}{n_1} +\frac{1}{n_2} )} }](https://tex.z-dn.net/?f=Z_%7BH_0%7D%3D%20%5Cfrac%7B%28p%27_1-p%27_2%29-%28p_1-p_2%29%7D%7B%5Csqrt%7Bp%27%281-p%27%29%2A%28%5Cfrac%7B1%7D%7Bn_1%7D%20%2B%5Cfrac%7B1%7D%7Bn_2%7D%20%29%7D%20%7D)
![p'= \frac{(p'_1+p'_2)}{2} = 0.45](https://tex.z-dn.net/?f=p%27%3D%20%5Cfrac%7B%28p%27_1%2Bp%27_2%29%7D%7B2%7D%20%3D%200.45)
![Z_{H_0}= \frac{(0.40-0.50)-0}{\sqrt{0.45*0.55(\frac{1}{225} +\frac{1}{276} )} } = -2.24](https://tex.z-dn.net/?f=Z_%7BH_0%7D%3D%20%5Cfrac%7B%280.40-0.50%29-0%7D%7B%5Csqrt%7B0.45%2A0.55%28%5Cfrac%7B1%7D%7B225%7D%20%2B%5Cfrac%7B1%7D%7B276%7D%20%29%7D%20%7D%20%3D%20-2.24)
Since the obtained value is less than the critical value the decision is to reject the null hypothesis.
At a level of 5%, you can say that the population proportion of women that purchased online past month is greater than the population proportion of men that purchased online past month.
I hope it helps!