Two independent samples yield: n1 = 25, x̄1 = 60, s1 = 9; n2 = 30, x̄2 = 55, s2 = 10. Construct a 90% confidence interval for μ1 - μ2 (use t* = 1.711 for df = 24).

How are hypothesis tests and confidence intervals related in the context of comparing two means?

Consider the following hypothesis test scenario. Assume random, independent samples from normally distributed populations. Find the critical values for (i) equal variances and (ii) not equal variances.

$H_a:{\mu }_1>{\mu }_2$, $\alpha =0.025$, $n_1=14$, $n_2=20$

A university administrator claims that the median number of years that faculty have worked at the university is less than the national median of $7.8$ years, based on a survey of the faculty at the university. The national median comes from a comprehensive higher education report. Is it possible that the administrator's claim is valid? What questions should you ask about the survey?

The mean SAT Math score for a sample of $100$ students is $540$, with a population standard deviation of $60$. The mean SAT Writing score for a sample of $130$ students is $555$, with a population standard deviation of $58$. At $\alpha =0.05$, can you support the claim that SAT Writing scores are higher than SAT Math scores?

A nutritionist believes that the mean daily calorie intake is higher for athletes than for non-athletes. In a sample of $16$ athletes, the mean intake is $3,150$ calories with a standard deviation of $410$. In a sample of $20$ non-athletes, the mean intake is $2,900$ calories with a standard deviation of $350$. At $\alpha =0.05$, is there enough evidence to support the nutritionist's belief? Assume population variances are not equal.

To evaluate battery life between two types of wireless headphones, a technician tests $20$ units of Brand A and $25$ units of Brand B.

Brand A has a mean battery life of $18.2$ $\text{hours}$ with a known population standard deviation of $1.5$ $\text{hours}$.

Brand B has a mean battery life of $17.5$ $\text{hours}$ with a known population standard deviation of $1.8$ $\text{hours}$.

At the $\alpha =0.05$ level of significance, can the technician conclude that there is a significant difference in the average battery life between the two brands?

How can you test a hypothesis about the difference between two independent population means with known standard deviations without using rejection regions?

Why is it important that the two samples in a two-sample t-test are independent?

Which of the following is NOT an assumption required for a valid two-sample t-test with unknown and unequal variances?