https://r2.erweima.ai/mathbot/image/1721680850695pdpk19n0.png

Let's start by solving Part 1 of the problem.

We need to set up the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)).

Given the context:
- The average number of Facebook friends people believe to be is \(140\).
- The sample average number of Facebook friends is \(146\).
- We want to test if the mean number of friends is greater than \(140\).

Thus, our hypotheses are:
- \(H_0\): The mean number of Facebook friends is less than or equal to \(140\). (\(\mu \leq 140\))
- \(H_1\): The mean number of Facebook friends is greater than \(140\). (\(\mu > 140\))

Next, identify the type of test. This is a right-tailed test because we are testing if the mean is greater than a specific value.

So, the hypotheses can be stated as:
- \(H_0: \mu \leq 140\)
- \(H_1: \mu > 140\)

Let's move on to the subsequent parts to solve the hypothesis test using the given dataLet's start by solving Part 1 of the problem.

We need to set up the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)).

Given the context:
- The average number of Facebook friends people believe to be is \(140\).
- The sample average number of Facebook friends is \(146\).
- We want to test if the mean number of friends is greater than \(140\).

Thus, our hypotheses are:
- \(H_0\): The mean number of Facebook friends is less than or equal to \(140\). (\(\mu \leq 140\))
- \(H_1\): The mean number of Facebook friends is greater than \(140\). (\(\mu > 140\))

Next, identify the type of test. This is a right-tailed test because we are testing if the mean is greater than a specific value.

So, the hypotheses can be stated as:
- \(H_0: \mu \leq 140\)
- \(H_1: \mu > 140\)

Let's move on to the subsequent parts to solve the hypothesis test using the given 

1. Sample mean (\(\bar{x}\)) = 146
2. Population standard deviation (\(\sigma\)) = 36.4
3. Sample size (n) = 32
4. Significance level (\(\alpha\)) = 0.10

We will calculate the test statistic and compare it with the critical value.

**Step 1: Calculate the test statistic \(z\)**:
\[ z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} \]

Where:
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.

**Step 2: Find the critical value for \(\alpha = 0.10\)** using the standard normal distribution table (z-table).

**Step 3: Compare the test statistic with the critical value to make a decision.**

Let's calculate it step by step.

1. **Calculate the test statistic \(z\)**:
   \[ z = \frac{146 - 140}{\frac{36.4}{\sqrt{32}}} \]
   \[ z = \frac{6}{\frac{36.4}{\sqrt{32}}} \]
   \[ z = \frac{6}{\frac{36.4}{5.656}} \]
   \[ z = \frac{6}{6.435} \]
   \[ z \approx 0.93 \]

2. **Find the critical value for \(\alpha = 0.10\)**:
   The critical value \(z_{\alpha}\) for a right-tailed test at \(\alpha = 0.10\) is approximately 1.28.

3. **Decision**:
   Since \(0.93 < 1.28\), we fail to reject the null hypothesis. There is not enough evidence to conclude that the mean number of Facebook friends is greater than 140 at the 0.10 significance level.

Would you like further details or have any questions?

Here are some additional questions you might have:

1. How to interpret the test results?
2. What if the significance level (\(\alpha\)) was different?
3. How to calculate p-value for this test?
4. Can we use t-test instead of z-test in this scenario?
5. What are the assumptions for performing this hypothesis test?

**Tip:** Always check the assumptions before performing hypothesis tests to ensure the validity of your results.

Learn how to solve a hypothesis testing problem regarding the mean number of Facebook friends. Understand concepts like null hypothesis, alternative hypothesis, and Z-test. Step-by-step calculation and interpretation provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation