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 - The
**alternative hypothesis** is - 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 e**nough 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.

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**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.

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**Proportion 1: **Jet-ski users

86.5% out of 400, thus:

**Proportion 2: **boaters

92.8% out of 250, so:

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**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

At the **alternative hypothesis**, it is tested that the **proportions are different**, that is, the** subtraction is different of 0.** So

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**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

From the **samples:**

The **value **of the **test statistic** is:

The **value **of the** test statistic** is **z = -2.67.**

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**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 e**nough 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