Hypothesis Testing Vocabulary

Hypothesis Testing

• Inference includes estimation of confidence intervals and hypothesis testing.

Big Picture of Hypothesis Testing

• Population: A statement is made about a population (e.g., average age is 50).
• Sample: A random sample is selected from the population.
• Sample Mean: The sample mean is calculated (e.g., 25).
• Comparison: Compare the sample statistic to the statement about the population.
  • If the sample statistic is close to the statement, the hypothesis is likely true.
  • If the sample statistic is far from the statement, the hypothesis is rejected.

Null and Alternative Hypotheses

• All hypothesis tests start with the null (H0) and alternative hypotheses.
• Statistical packages like Stata require the user to specify these hypotheses correctly.
• Null Hypothesis (H0): A status quo hypothesis; something that exists unless evidence suggests otherwise.
• Alternative Hypothesis: Everything that is not the null hypothesis.

Procedure to Identify Hypotheses

1. State the Question: E.g., The average age of the population (\mu) is equal to 50. \mu = 50
2. State the Opposite: The opposite of \mu = 50 is \mu \neq 50.
3. Identify the Alternative: The alternative hypothesis cannot have an equal sign.
   • Alternative can have only strictly greater ($>$), strictly less ($<$), or not equal (\neq) signs.
4. State the Null (H0): The remaining statement becomes the null hypothesis.

Types of Tests

• Two-Sided Test (Two-Tailed Test): Alternative hypothesis is "not equal to".
  • Null hypothesis: \mu = \mu_0
• One-Sided Test: Alternative hypothesis has a direction (strictly greater or strictly less).
  • Left-Sided Test: Null hypothesis could be \mu \le \mu0, and the alternative is \mu > \mu0.
  • Right-Sided Test: Null hypothesis could be \mu \ge \mu0, and the alternative is \mu < \mu0.
• Null and alternative hypotheses should be mutually exclusive and exhaustive.

Importance of the Alternative Hypothesis

• The alternative hypothesis is crucial and influences many parts of the test.

Example Question 1

• The mean score is greater than or equal to 80.
  1. \mu \ge 80
  2. Opposite: \mu < 80
  3. Alternative: \mu < 80
  4. Null: \mu \ge 80

Example Question 2

• The mean score is less than 80.
  1. \mu < 80
  2. Opposite: \mu \ge 80
  3. Alternative: \mu < 80
  4. Null: \mu \ge 80

Determining One-Sided vs. Two-Sided Test

• Based on the alternative hypothesis.
• One-sided: Alternative points left ($
• Two-sided: Alternative states the value could be either too high or too low (\neq).

Example: Average Earnings

• Test the average earnings of $40,000.
  1. \mu = 40,000
  2. Opposite: \mu \neq 40,000
  3. Alternative: \mu \neq 40,000 (Two-sided)
  4. Null: \mu = 40,000
• If the question indicates a direction (greater, less, smaller, larger), it is a one-sided test.
• If the question indicates equal or not equal, it is a two-sided test.

Significance Level

• Denoted by \alpha
• Alpha (\alpha) is the probability of a Type I error.
• Type I error: Rejecting H0 when H0 is actually true.
• Typical alpha values: 0.05, 0.01, 0.10 (corresponding to 95%, 99%, and 90% confidence intervals).

Type I and Type II Errors

• We choose a small \alpha to minimize the chance of making a Type I error.

Test Statistic

• A sample statistic that is standardized to determine how far away it is from the null hypothesis.
• If the test statistic is far from the null, we reject the null.
• Denoted as T.
• T distribution gets very close to Z distribution when the sample size is large.

Calculating the T Statistic

• Standardize the sample mean: T = \frac{\bar{X} - \mu_0}{SE(\bar{X})}
  • \bar{X} = sample mean
  • \mu_0 = value under the null hypothesis
  • SE(\bar{X}) = standard error of the sample mean
• This test statistic has a T distribution with n-1 degrees of freedom.

Example Calculation

• Sample size (n) = 171
• Sample mean = 41413
• Standard error of the mean = 1952
• Test hypothesis: Average earnings are 40,000.

T = \frac{41413 - 40000}{1952} = 0.724

• T distribution with 170 degrees of freedom.

Approaches to State the Test Decision

1. P-value approach
2. Rejection-regions (critical-regions) approach

P-Value Approach

• How likely is it to obtain a test statistic of 0.724 or greater from the T distribution?
• P-value: The probability of obtaining a test statistic at least as large in absolute value as the one observed.
• Formula: P-value = P(|T_{n-1}| > t)

Calculating P-Value for Two-Sided Tests

1. Sketch the T distribution.
2. Place the value of the test statistic on the graph.
3. Shade the area in the tails greater than the absolute value of the test statistic.

• Alternative approach: P-value is twice the area of the tail in the direction of the sign of T.

Calculating P-Value for One-Tailed Tests

• Go in the direction of the alternative.
• If the alternative is \mu > \mu_0, shade the area to the right of the test statistic.
• If the alternative is \mu < \mu_0, shade the area to the left of the test statistic.

Using the P-Value

• Compare the P-value to the level of significance \alpha.
• If P-value < \alpha, reject H0.
• If P-value > \alpha, fail to reject H0.

Example

• P-value = 0.47
• \alpha = 0.05
• P-value > \alpha, so fail to reject H0.

Critical Regions Approach

• Compare the test statistic to the critical value.
• Critical regions: Range of values for the test statistic for which H0 will be rejected.
• If the test statistic falls in the rejection region, reject H0.

Determining Rejection Region

• For two-sided tests, split \alpha into two tails (\alpha/2 in each tail).
• For one-tailed tests, the entire \alpha is in one tail.

Example

• Two-sided test at \alpha = 0.05: \alpha/2 = 0.025 in each tail.
• Critical value from t-table: c = 1.974.

Decision Rule

• If absolute value of test statistic > C, reject H0.
• If absolute value of test statistic < C, fail to reject H0.
• In the example, the test statistic was 0.724, which is not in the rejection region, so fail to reject.
  Note: Both approaches (P-value and critical regions) should lead to the same decision.

Hypothesis Testing in Stata

Confidence Interval

• Command: mean earnings
  • Provides the mean, standard error, and 95% confidence interval.
  • The numbers are also available by using return list
• To specify a different confidence level, use the level option, e.g., mean earnings, level(99).

Calculating the Test Statistic

• Use the display command as a calculator.
  • display (41412.69 - 40000) / 1952.103

Calculating the P-Value

• Use the ttail command to get the probability from the T distribution.
  • help ttail to learn the syntax.
  • display ttail(170, 0.724) * 2 (for a two-tailed test).

T-Test Command in Stata

• Command: ttest earnings == 40000
  • Specifies the null hypothesis as equality.
  • Stata provides information about the test, including the sample size, mean, standard deviation, standard error, and confidence interval.
  • Stata performs all three tests (left-sided, right-sided, and two-sided) and provides the corresponding p-values.
  • Always look at the output and ensure the test is what you want to use.
  • Verify that Stata has calculated sample size, sample mean, and standard deviation as you expect.
  • If you cannot generate the same numbers it is an indication you need more help before attempting the test.
  • The mean, for two sample tests can be calculated differently, so make sure you know what the mean represents.

Practice Problems

Gas Prices

• Average price for all California gas stations: 3.81
• Test that the average price in Yolo County is neither higher nor lower than in California.
• Question: One-sided or two-sided test?
• Answer: Two-sided test (because it could be higher or lower, i.e., not equal).

Steps

1. Question: \mu \neq 3.81
2. Opposite: \mu = 3.81
3. Alternative: \mu \neq 3.81
4. Null: \mu = 3.81

P Value Approach

• P value is the probability that T is greater than 5.256 times 2
• So the numbers are close to zero, so you reject H0 at alpha = 0.05
• Found the evidence that the average gas prices in your account were not $3.81 at that point in time.