From the test the person wants, and the sample data, we build the test hypothesis, find the test statistic, and use this to reach a conclusion.
This is a two-sample test, thus, it is needed to understand the central limit theorem and subtraction of normal variables.
Doing this:
- The null hypothesis is
![H_0: p_1 - p_2 = 0 \rightarrow p_1 = p_2](https://tex.z-dn.net/?f=H_0%3A%20p_1%20-%20p_2%20%3D%200%20%5Crightarrow%20p_1%20%3D%20p_2)
- The alternative hypothesis is
![H_1: p_1 - p_2 \neq 0 \rightarrow p_1 \neq p_2](https://tex.z-dn.net/?f=H_1%3A%20p_1%20-%20p_2%20%5Cneq%200%20%5Crightarrow%20p_1%20%5Cneq%20p_2)
- The value of the test statistic is z = -2.67.
- The p-value of the test is 0.0076 < 0.05(standard significance level), which means that there is enough evidence to conclude that the proportion of people who wear life vests while riding a jet ski is not the same as the proportion of people who wear life vests while riding in a boat.
-------------------
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
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.
-------------------------------------
Proportion 1: Jet-ski users
86.5% out of 400, thus:
![p_1 = 0.865](https://tex.z-dn.net/?f=p_1%20%3D%200.865)
![s_1 = \sqrt{\frac{0.865*0.135}{400}} = 0.0171](https://tex.z-dn.net/?f=s_1%20%3D%20%5Csqrt%7B%5Cfrac%7B0.865%2A0.135%7D%7B400%7D%7D%20%3D%200.0171)
Proportion 2: boaters
92.8% out of 250, so:
![p_2 = 0.928](https://tex.z-dn.net/?f=p_2%20%3D%200.928)
![s_2 = \sqrt{\frac{0.928*0.072}{250}} = 0.0163](https://tex.z-dn.net/?f=s_2%20%3D%20%5Csqrt%7B%5Cfrac%7B0.928%2A0.072%7D%7B250%7D%7D%20%3D%200.0163)
------------------------------------------------
Hypothesis:
Test the claim that the proportion of people who wear life vests while riding a jet ski is not the same as the proportion of people who wear life vests while riding in a boat.
At the null hypothesis, it is tested that the proportions are the same, that is, the subtraction is 0. So
![H_0: p_1 - p_2 = 0 \rightarrow p_1 = p_2](https://tex.z-dn.net/?f=H_0%3A%20p_1%20-%20p_2%20%3D%200%20%5Crightarrow%20p_1%20%3D%20p_2)
At the alternative hypothesis, it is tested that the proportions are different, that is, the subtraction is different of 0. So
![H_1: p_1 - p_2 \neq 0 \rightarrow p_1 \neq p_2](https://tex.z-dn.net/?f=H_1%3A%20p_1%20-%20p_2%20%5Cneq%200%20%5Crightarrow%20p_1%20%5Cneq%20p_2)
------------------------------------------------------
Test statistic:
The test statistic is:
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 samples:
![X = p_1 - p_2 = 0.865 - 0.928 = -0.063](https://tex.z-dn.net/?f=X%20%3D%20p_1%20-%20p_2%20%3D%200.865%20-%200.928%20%3D%20-0.063)
![s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.0171^2 + 0.0163^2} = 0.0236](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bs_1%5E2%20%2B%20s_2%5E2%7D%20%3D%20%5Csqrt%7B0.0171%5E2%20%2B%200.0163%5E2%7D%20%3D%200.0236)
The value of 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)
![z = \frac{-0.063 - 0}{0.0236}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B-0.063%20-%200%7D%7B0.0236%7D)
![z = -2.67](https://tex.z-dn.net/?f=z%20%3D%20-2.67)
The value of the test statistic is z = -2.67.
---------------------------------------------
p-value of the test and decision:
The p-value of the test is the probability that the proportion differs by at at least 0.063, which is P(|z| > 2.67), given by 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(standard significance level), which means that there is enough evidence to conclude that the proportion of people who wear life vests while riding a jet ski is not the same as the proportion of people who wear life vests while riding in a boat.
A similar question is found at brainly.com/question/24250158