Determine the null hypothesis, H Subscript 0, and the alternative hypothesis, H Subscript 1. Let males be population 1 and females be population 2. H Subscript 0: mu 1 minus mu 2 equals 0 H Subscript 1: mu 1 minus mu 2 not equals 0 (Type integers or decimals. Do not round.) Part 2 Compute the test statistic.
The Correct Answer and Explanation is:
Let’s break down the problem step by step.
Null and Alternative Hypotheses
In hypothesis testing, we compare two populations (in this case, males and females), which are represented by Population 1 (males) and Population 2 (females). The null hypothesis (H₀) and the alternative hypothesis (H₁) are statements that we test in a statistical procedure.
- Null Hypothesis (H₀): This is a statement of no difference between the two populations. It assumes that the difference in the population means is zero. H0:μ1−μ2=0H_0: \mu_1 – \mu_2 = 0H0:μ1−μ2=0 Here, μ1\mu_1μ1 is the mean of the population of males, and μ2\mu_2μ2 is the mean of the population of females. The null hypothesis suggests that there is no difference in the means of the two populations.
- Alternative Hypothesis (H₁): This is the opposite of the null hypothesis, stating that there is a difference between the two population means. This hypothesis could indicate either an increase or a decrease, and is typically expressed as a two-tailed test. H1:μ1−μ2≠0H_1: \mu_1 – \mu_2 \neq 0H1:μ1−μ2=0 This indicates that the difference between the means is not zero, i.e., the means of males and females are significantly different.
Test Statistic
To compute the test statistic, we often use a t-test or z-test depending on the sample size and whether the population standard deviations are known. The formula for the test statistic when comparing two sample means (assuming equal variances) is: t=Xˉ1−Xˉ2s12n1+s22n2t = \frac{\bar{X}_1 – \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}t=n1s12+n2s22Xˉ1−Xˉ2
Where:
- Xˉ1\bar{X}_1Xˉ1 and Xˉ2\bar{X}_2Xˉ2 are the sample means of the two populations (males and females).
- s1s_1s1 and s2s_2s2 are the sample standard deviations of the two populations.
- n1n_1n1 and n2n_2n2 are the sample sizes of the two populations.
Steps to Compute the Test Statistic:
- Determine the sample means Xˉ1\bar{X}_1Xˉ1 and Xˉ2\bar{X}_2Xˉ2 for males and females, respectively.
- Find the sample standard deviations s1s_1s1 and s2s_2s2 for each population.
- Calculate the sample sizes n1n_1n1 and n2n_2n2 for each population.
- Substitute these values into the formula for the test statistic to get the value of ttt.
Example:
If we had sample data like this:
- Males: Xˉ1=75\bar{X}_1 = 75Xˉ1=75, s1=10s_1 = 10s1=10, n1=30n_1 = 30n1=30
- Females: Xˉ2=70\bar{X}_2 = 70Xˉ2=70, s2=12s_2 = 12s2=12, n2=30n_2 = 30n2=30
Substitute into the formula: t=75−7010230+12230t = \frac{75 – 70}{\sqrt{\frac{10^2}{30} + \frac{12^2}{30}}}t=30102+3012275−70 t=510030+14430t = \frac{5}{\sqrt{\frac{100}{30} + \frac{144}{30}}}t=30100+301445 t=53.33+4.8t = \frac{5}{\sqrt{3.33 + 4.8}}t=3.33+4.85 t=58.13t = \frac{5}{\sqrt{8.13}}t=8.135 t=52.85≈1.75t = \frac{5}{2.85} \approx 1.75t=2.855≈1.75
Thus, the test statistic would be approximately 1.75.
Interpretation:
- Once you calculate the test statistic, you would compare it to the critical value from the t-distribution (for a t-test) or z-distribution (for a z-test) at a specific significance level (e.g., 0.05).
- If the absolute value of the test statistic is greater than the critical value, you reject the null hypothesis. If it’s smaller, you fail to reject the null hypothesis.
