Hypothesis Testing Explained

Hypothesis Testing Overview

• Three methods to test hypotheses are: the traditional method, the P-value method, and the confidence interval method.

Hypothesis Testing - Traditional Method

• A statistical hypothesis is an unproven conjecture about a population parameter that may or may not be true.
• The null hypothesis, denoted as H₀, is a statistical hypothesis stating no difference between a parameter and a specific value or between two parameters.
• The alternative hypothesis, denoted as H₁, is a statistical hypothesis asserting a difference between a parameter and a specific value, or between two parameters.

Types of Hypothesis Tests

• Situation A involves a medical researcher testing if a new medication has side effects on pulse rate, with a null hypothesis of μ = 82 and an alternative hypothesis of μ ≠ 82; this is a two-tailed test.
• Situation B involves a chemist testing an additive to increase battery life, with a null hypothesis of μ = 36 months and an alternative hypothesis of μ > 36 months; this is a right-tailed test.
• Situation C involves a contractor testing insulation to lower heating bills, with a null hypothesis of μ = $78 and an alternative hypothesis of μ < $78; this is a left-tailed test.

Claims in Hypothesis Testing

• Researchers generally seek evidence to support a claim, stated as the alternative or research hypothesis.
• Evidence can only support the claim if it's the alternative hypothesis.
• Evidence can reject the claim if it's the null hypothesis.

Designing the Hypothesis Test

• Designing the study involves choosing the correct statistical test, significance level, and formulating a plan.
• A statistical test uses sample data to decide whether the null hypothesis should be rejected.
• The test value is the numerical value from a statistical test.
• Four possible outcomes exist in hypothesis testing.

Type I and II Errors

• A Type I error occurs when the null hypothesis is rejected when it is true.
• A Type II error occurs when the null hypothesis is not rejected when it is false.
• The level of significance is the maximum probability of committing a Type I error, symbolized by α, where P(Type I error) = α.
• The probability of a Type II error is symbolized by β, where P(Type II error) = β.
• Typical significance levels are 0.10, 0.05, and 0.01. If α = 0.10, there is a 10% chance of rejecting a true null hypothesis.

Critical and Nonrejection Regions

• The critical or rejection region is the range of test values that indicates a significant difference, leading to the rejection of the null hypothesis.
• The noncritical or nonrejection region contains test values, suggesting differences, likely due to chance, and the null hypothesis should not be rejected.
• The critical value (C.V.) separates the critical region from the noncritical region.
• For a right-tailed test with α = 0.01, z = 2.33.
• For a left-tailed test with α = 0.01, z = -2.33 due to symmetry.
• For a two-tailed test with α = 0.01, z = ±2.58.
• Finding the critical values for specific α values using Table E involves drawing a figure to indicate the area:
• For left-tailed tests, the critical region with area α is on the left side of the mean.
• For right-tailed tests, the critical region with area α is on the right side of the mean.
• For two-tailed tests, α is divided by 2, with half to the right and half to the left of the mean.
• For a left-tailed test, use the z value corresponding to the area equivalent to α in Table E.
• For a right-tailed test, use the z value corresponding to the area equivalent to 1 - α.
• For a two-tailed test, use the z value corresponding to α/2 for the left value (negative) and the z value corresponding to the area equivalent to 1 – α/2 for the right value (positive).

Steps for Solving Hypothesis-Testing Problems (Traditional Method)

• State the hypotheses and identify the claim.
• Find the critical value(s) from the appropriate table.
• Compute the test value.
• Make the decision to reject or not reject the null hypothesis.
• Summarize the results.

z Test for a Mean

• The z test is a statistical test for the mean of a population, usable when n ≥ 30, or when the population is normally distributed and σ is known.

Example 8-3: Intelligence Tests

• Pennsylvania: average IQ score is 101.5; population standard deviation is 15.
• A school claims their students have higher IQ scores.
• A random sample of 30 students has a mean of 106.4. Test the claim at α = 0.05.
• Hypotheses: H₀: μ = 101.5 and H₁: μ > 101.5 (claim).
• Critical value z = +1.65.
• Decision: Reject the null hypothesis because the test value, 1.79, is greater than the critical value, 1.65.
• There is enough evidence to support the claim that student IQ is higher than the state average IQ.
• The difference is statistically significant with a maximum Type I error probability of 0.05 or 5%.

Example 8-4: SAT Tests

• Average SAT math test is 515.
• Population standard deviation is 100.
• Superintendent wants to see if students scored below the national average, selecting 36 scores.
• Alpha is 0.10.
• Hypotheses: H₀: μ = 515 and H₁: μ < 515 (claim).
• Critical value z = −1.28.
• The mean of the data = 509.028.
• The test value, -0.36, falls in the noncritical region, so the decision is to not reject the null hypothesis.
• There is not enough evidence to support the claim that the students scored below the national average.

Example 8-5: Cost of Rehabilitation

• Average cost of stroke rehabilitation: $24,672 (foundation report).
• Researcher selects 35 stroke victims, average cost of rehabilitation is $26,343; population standard deviation is $3251.
• Alpha is 0.01.
• Hypotheses: H₀: μ = $24,672 and H₁: μ ≠ $24,672 (claim).
• Critical values are z = ±2.58.
• Reject the null hypothesis because the test value falls in the critical region.
• There is enough evidence to support the claim that the average rehabilitation cost at the particular hospital differs from $24,672.

P-Value Method

• The P-value (or probability value) represents the likelihood of obtaining a sample statistic like the mean or a more extreme value in the direction of the alternative hypothesis, assuming the null hypothesis is true.
• If the P-value is less than or equal to alpha, reject the null hypothesis.
• If the P-value is greater than alpha, fail to reject the null hypothesis.

Steps for Solving Hypothesis-Testing Problems (P-Value Method)

• State the hypotheses and identify the claim.
• Compute the test value.
• Find the P-value.
• Make the decision.
• Summarize the results.

Example 8-6: Cost of College Tuition

• Average tuition and fees at a four-year public college is greater than $5700.
• A random sample of 36 four-year public colleges is $5950.
• Population standard deviation is $659. Alpha is 0.05.
• Hypotheses: H₀: μ = $5700 and H₁: μ > $5700 (claim).
• Since the P-value is less than 0.05, the decision is to reject the null hypothesis.
• There is enough evidence to support the claim that the tuition and fees at four-year public colleges are greater than $5700.
• If α = 0.01, the null hypothesis would not be rejected.

Example 8-7: Wind Speed

• Claim: The average wind speed in a certain city is 8 miles per hour.
• A sample of 32 days has an average wind speed of 8.2 miles per hour.
• Standard deviation of the population is 0.6 mile per hour and α = 0.05.
• Hypotheses: H₀: μ = 8 (claim) and H₁: μ ≠ 8.
• The area for z = 1.89 is 0.9706.
• The P-value is 2(0.0294) = 0.0588
• It is recommended do not not reject the null hypothesis because the P-value is greater than 0.05.

P-Value Guidelines

• If P-value ≤ 0.01, reject the null hypothesis. The difference is highly significant.
• If P-value > 0.01 but P-value ≤ 0.05, reject the null hypothesis. The difference is significant.
• If P-value > 0.05 but P-value ≤ 0.10, consider the consequences of Type I error before rejecting the null hypothesis.
• If P-value > 0.10, do not reject the null hypothesis. The difference is not significant.

Statistical vs Practical Significance

• Distinguishing between statistical and practical significance is important.
• Rejecting the null hypothesis at a specific significance level means the difference is likely not due to chance and is statistically significant.
• Results may lack practical significance.
• Researchers should apply common sense when interpreting statistical test results.

T Test for a Mean

• A statistical test for the mean of a population is used when the population is normally or approximately normally distributed, and σ is unknown.
• Degrees of freedom are d.f. = n - 1.

Example 8-9 and 8-10: Table F

• For α = 0.01 with d.f. = 22 for a left-tailed test, the critical value is t = -2.508.
• For α = 0.10 with d.f. = 18 for a two-tailed test, the critical values are 1.734 and -1.734.

Example 8-12: Hospital Infections

• The claim states that the average number of infections per week at a hospital in southwestern Pennsylvania is 16.3.
• Ten weeks had a mean number of 17.7 infections, with a sample standard deviation of 1.8 and α = 0.05 and d.f. = 9.
• Hypotheses: H₀: μ = 16.3 (claim) and H₁: μ ≠ 16.3.
• The critical values are 2.262 and -2.262.
• There is sufficient evidence to reject the claim that the average number of infections is 16.3.

Example 8-13: Starting Nurse Salary

• Beginning salary is $79,500. The goal is to test the salary is less than $79,500 and α = 0.10.
• The salaries given are variable normally distributed.
• Hypotheses: H₀: μ = $79,500 and H₁: μ < $79,500 (claim)
• The critical value is -1.415.
• Since -0.773 falls in the critical region then there is evidence to support the claim that the average salary is less than $79,500.