Hypothesis Testing - Z test & T test

Summary

This video explains the difference between the Z test and the T test in hypothesis testing, highlighting when to use each based on sample size and population variance. The Z test is used for sample sizes greater than 30 or known population variance, while the T test is applied for smaller samples or unknown variance. Through two example exercises, the script walks viewers through formulating null and alternative hypotheses, determining rejection regions, and calculating Z and T values to test hypotheses. The focus is on understanding when to reject or fail to reject the null hypothesis at different confidence levels.

Takeaways

• 😀 Z-test is used when the population variance is known or the sample size is larger than 30.
• 😀 T-test is used when the sample size is less than 30 and the population variance is unknown.
• 😀 The most common factor for choosing between the Z-test and T-test is the sample size.
• 😀 A Z-test is used when the sample size is 100 or more (n ≥ 30).
• 😀 In hypothesis testing, the null hypothesis (H₀) represents the traditionally accepted view, and the alternative hypothesis (H₁) challenges it.
• 😀 A two-tailed test is used when the alternative hypothesis suggests that a parameter is different from a specific value, without specifying a direction.
• 😀 In the Z-test example, the null hypothesis for average lifetime is that it's 75 years, and the alternative hypothesis is that it's different from 75 years.
• 😀 The critical value for a 95% confidence level in a Z-test is ±1.96, which defines the rejection regions for the hypothesis test.
• 😀 In the T-test example, the null hypothesis states that the price of gasoline is $2.45, and the alternative hypothesis suggests it’s higher.
• 😀 The critical value for a T-test is based on the significance level (α), the sample size, and the degrees of freedom, which in this case was 2.49 for 24 degrees of freedom at 99% confidence.
• 😀 The decision to reject or fail to reject the null hypothesis is based on comparing the test statistic (Z or T value) to the critical value from the appropriate table.
• 😀 A test statistic value that falls within the rejection region leads to rejecting the null hypothesis, indicating a significant result.

Q & A

What is the primary factor for deciding whether to use a Z test or a T test in hypothesis testing?

The primary factor is the sample size. If the sample size is 30 or larger, the Z test is used. If the sample size is smaller than 30, the T test is used.

When is the Z test appropriate, and when should the T test be used?

The Z test is appropriate when the population variance is known or when the sample size is larger than 30. The T test should be used when the sample size is smaller than 30 and the population variance is unknown.

In the first example, what is the null hypothesis regarding the average lifetime of females in the US?

The null hypothesis is that the average lifetime of females in the US is 75 years.

Why is the test in the first example a two-tailed test?

The test is two-tailed because the alternative hypothesis states that the average lifetime could be either higher or lower than 75 years.

How are the rejection regions determined for the Z test?

The rejection regions are determined based on the significance level. In this case, at a 95% confidence level, the rejection regions are 2.5% on both sides of the distribution, corresponding to Z values of ±1.96.

What is the formula for calculating the Z value in the first example, and what is the result?

The formula for calculating the Z value is: Z = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size)). Plugging in the values, the Z value is approximately 1.43.

What conclusion is made from the Z test in the first example?

Since the Z value (1.43) falls outside the rejection region (±1.96), we fail to reject the null hypothesis, meaning there is not enough evidence to say that the average lifetime of females is different from 75 years.

In the second example, why is a T test used instead of a Z test?

A T test is used because the sample size is 25, which is less than 30, and the population variance is unknown.

What is the significance level used in the second example, and what is the critical value for the T test?

The significance level is 0.01 (for a 99% confidence level), and the critical value for the T test with 24 degrees of freedom is approximately 2.49.

What does the result of the T test in the second example imply about gasoline prices?

The T value of 2.86 exceeds the critical value of 2.49, so we reject the null hypothesis. This implies that the average price of gasoline in the Midwest is indeed higher than $2.45.