Math Problem Solver

a. One-Sided and Two-Sided Hypothesis

A one-sided hypothesis test is used when the alternative hypothesis specifies a directional change, such as H_1: \mu > \mu_0 or H_1: \mu < \mu_0 . This indicates interest only in increases or decreases in the parameter being tested. In contrast, a two-sided hypothesis test assesses whether the parameter is significantly different from the null value in either direction, such as H_1: \mu \neq \mu_0 . For instance, testing whether a drug increases or decreases blood pressure would require a two-sided test, while testing whether it only increases blood pressure would require a one-sided test. The choice of test depends on the research question, and a one-sided test typically has greater power to detect an effect in the specified direction but does not account for changes in the opposite direction.

Example: A factory claims that the mean weight of its product is 50 kg. A sample of 25 products has a mean weight of 51 kg and a standard deviation of 2 kg. Test whether the weight is greater than 50 kg at a significance level of 0.05 (one-sided test).

• Null Hypothesis (H₀): \mu = 50

• Alternative Hypothesis (H₁): \mu > 50 (one-sided)

• Significance Level: \alpha = 0.05

1. Calculate the test statistic (t):

t = \frac{x̄ - \mu}{s / \sqrt{n}} = \frac{51 - 50}{2 / \sqrt{25}} = \frac{1}{0.4} = 2.5

2. Find the critical value: For df = 24 and \alpha = 0.05 , t_{critical} = 1.711 .

3. Decision: Since t = 2.5 > t_{critical} = 1.711 , reject H₀ . The mean weight is significantly greater than 50 kg.

b. P-Value in Hypothesis Test

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis: the smaller the p-value, the stronger the evidence to reject H_0 . If the p-value is less than the significance level ( \alpha ), we reject H_0 ; otherwise, we fail to reject it. The p-value approach is widely used because it provides an exact measure of evidence and eliminates reliance on pre-set critical values. For example, a p-value of 0.03 indicates a 3% chance of observing the sample data if H_0 were true.

Example: Using the same factory data ( x̄ = 51 , s = 2 , n = 25 ), calculate the p-value for the one-sided hypothesis H₁: \mu > 50 .

1. Test statistic (t): Already calculated as t = 2.5 .

2. Degrees of freedom (df): n - 1 = 24 .

3. Find the p-value: Using a t-distribution table or software, p(t > 2.5) \approx 0.01 .

4. Decision: Since p = 0.01 < \alpha = 0.05 , reject H₀ . The mean weight is significantly greater than 50 kg.

c. General Procedure for Hypothesis Testing

Hypothesis testing follows a structured approach. First, define the null hypothesis ( H_0 ) and alternative hypothesis ( H_1 ). Next, choose the significance level ( \alpha , typically 0.05) and select the appropriate test based on the data and sample conditions (e.g., z-test, t-test). Then, calculate the test statistic using the sample data and compare it to the critical value from the relevant distribution or compute the p-value. Finally, make a decision: reject H_0 if the test statistic exceeds the critical value or if the p-value is less than \alpha . Otherwise, fail to reject H_0 . This procedure ensures consistency and objectivity in decision-making.

Example: A company claims 60% of its customers are satisfied. In a survey of 200 customers, 130 are satisfied. Test whether the satisfaction rate differs from 60% at \alpha = 0.05 .

• Null Hypothesis (H₀): p = 0.60

• Alternative Hypothesis (H₁): p \neq 0.60 (two-sided)

• Sample proportion ( p̂ ): 130 / 200 = 0.65

1. Calculate the standard error (SE):

SE = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.60(1 - 0.60)}{200}} = 0.0346

2. Calculate the z-statistic:

z = \frac{p̂ - p}{SE} = \frac{0.65 - 0.60}{0.0346} = 1.45

3. Find the p-value: For z = 1.45 , the p-value is approximately 0.147.

4. Decision: Since p = 0.147 > 0.05 , fail to reject H₀ . There is insufficient evidence to conclude that the satisfaction rate differs from 60%.