Sofia is conducting a survey among her 10 friends to compare their average daily social media usage to the global average of 142 minutes. In order to determine whether their usage is significantly higher, she can perform a hypothesis test. Here's how she can approach it:

Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): The average daily usage of social media among Sofia's friends is the same as the global average. This means there is no significant difference.

  $H_0: \mu = 142$

• Alternative Hypothesis ($H_a$): The average daily usage of social media among Sofia's friends is significantly higher than the global average.

  $H_a: \mu > 142$

This is a one-tailed test because we're only interested in whether their usage is greater than the global average, not less.

Step 2: Choose the Significance Level

Typically, a significance level of $\alpha = 0.05$ is chosen, meaning Sofia will accept a 5% risk of making a Type I error (rejecting a true null hypothesis).

Step 3: Calculate the Test Statistic

If Sofia knows the sample mean ($\bar{x}$) of her friends' social media usage and the standard deviation (s), she can calculate the test statistic (t-value) using the t-test formula, since she is working with a small sample size (n = 10):

$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$

Where:

• $\bar{x}$ = sample mean (average usage of her friends),
• $\mu$ = population mean (142 minutes),
• $s$ = sample standard deviation,
• $n$ = sample size (10).

Step 4: Find the Critical Value

Using a t-distribution table, Sofia can find the critical t-value for $\alpha = 0.05$ and degrees of freedom (df = n - 1 = 9).

Step 5: Compare the Test Statistic to the Critical Value

• If the calculated t-value is greater than the critical t-value, then Sofia can reject the null hypothesis, concluding that her friends' average social media usage is significantly higher than the global average.
• If the t-value is smaller or equal to the critical value, Sofia fails to reject the null hypothesis.

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Would you like to proceed with calculating the test statistic if you have the data from Sofia’s survey? Or would you like to go over any of the steps in more detail?

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5 Related Questions:

1. What is the difference between a one-tailed and a two-tailed hypothesis test?
2. How do you calculate the sample standard deviation?
3. What are the implications of rejecting or failing to reject the null hypothesis?
4. How do you interpret a p-value in hypothesis testing?
5. What is the concept of degrees of freedom in t-tests?

Tip: In a small sample size, such as 10, it's important to use the t-distribution instead of the normal distribution, as the latter might not give accurate results.