Hypothesis Testing: Two-tailed z test for mean
Summary
TLDRThis 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.
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