Hypothesis Testing: Two-tailed z test for mean

Summary

This tutorial explores a hypothesis test to determine if the mean age of college students in city X has changed from 23. Using a sample mean of 23.8 with a standard deviation of 2.4, the video demonstrates a z-test at a 0.05 significance level, leading to the rejection of the null hypothesis and the conclusion that the mean age has changed. However, when the test is conducted at a 0.02 significance level, the null hypothesis is not rejected, indicating insufficient evidence for a change in the mean age. The video effectively illustrates the impact of different significance levels on hypothesis testing outcomes.

Takeaways

• 📚 The mean age of college students in city X has historically been 23, but a recent sample indicates a potential change.
• 🔍 A hypothesis test is being conducted to determine if there is evidence of a change in the mean age from 23 at a 5% significance level (α = 0.05).
• 🎯 The null hypothesis (H0) states that the mean age remains at 23, while the alternative hypothesis (H1) suggests it is not equal to 23.
• 📉 The test is two-tailed, dividing α by 2 to get 0.025 in each tail, indicating the rejection region for the test.
• 📊 A z-test is chosen for this analysis because the population standard deviation (σ) is known, and the sample size is sufficiently large.
• 🔢 The critical z-values for the rejection region are determined to be -1.96 and +1.96 using the standard normal distribution table.
• 🧐 The test statistic is calculated using the sample mean, population mean, standard deviation, and sample size, resulting in a z-score of 2.16.
• 🚫 At α = 0.05, the test statistic exceeds the critical value, leading to the rejection of the null hypothesis and supporting the claim that the mean age has changed.
• 🔄 When the significance level is lowered to 2% (α = 0.02), the critical z-values change to -2.33 and +2.33, indicating a more stringent test.
• 📉 The same test statistic of 2.16 does not fall into the new rejection region at α = 0.02, so the null hypothesis is not rejected, and there's insufficient evidence to claim a change in mean age.
• 📈 The tutorial demonstrates the impact of the chosen significance level on the outcome of a hypothesis test and the importance of correctly interpreting statistical results.

Q & A

What is the mean age of college students in city X historically?

Historically, the mean age of all college students in city X has been 23.

What was the mean age found in the random sample of 42 students this year?

The mean age found in the random sample of 42 students this year was 23.8.

What is the population standard deviation of the students' ages?

The population standard deviation of the students' ages is 2.4.

What is the null hypothesis in this scenario?

The null hypothesis is that the population mean age has not changed and is equal to 23.

What is the alternative hypothesis if we are testing for a change in the mean age?

The alternative hypothesis is that the population mean age is not equal to 23, indicating a change.

What is the significance level (α) used for the initial hypothesis test?

The significance level (α) used for the initial hypothesis test is 0.05.

Why is the z-test used in this example instead of the t-test?

The z-test is used in this example because the population standard deviation (σ) is known.

What are the critical z-values that define the rejection region for α = 0.05?

The critical z-values for α = 0.05 are -1.96 and +1.96, which define the rejection region.

What is the calculated test statistic for the hypothesis test?

The calculated test statistic for the hypothesis test is 2.16.

What conclusion can be drawn if the test statistic is 2.16 and α = 0.05?

If the test statistic is 2.16 and α = 0.05, we reject the null hypothesis, concluding there is enough evidence to infer that the mean age has changed.

What would be the critical z-value if the significance level is changed to α = 0.02?

If the significance level is changed to α = 0.02, the critical z-value would be approximately ±2.33.

What is the conclusion if the test is conducted at α = 0.02 with the same test statistic of 2.16?

At α = 0.02, with the same test statistic of 2.16, we fail to reject the null hypothesis, meaning there is not enough evidence to infer that the mean age has changed.