Test on Introduction to Statistics and Probability

Question 1: In the four parts of hypothesis testing, if the test statistic falls into the rejection region, the null hypothesis is rejected.

A. Yes

B. No

Explanation: The 'Decision' part of hypothesis testing states that if the test statistic falls into the rejection region, the null hypothesis will be rejected. Otherwise, it will not.

Question 2: When conducting a hypothesis test for the difference between two population means where the population standard deviations (σ1 and σ2) are unknown but assumed to be equal, which of the following statements is TRUE?

A. The test statistic follows a Z-distribution.

B. The null hypothesis typically states that μ1 = μ2.

C. The degrees of freedom for the t-distribution are calculated as n1 + n2 - 1.

D. Rejection of the null hypothesis in a two-tailed test occurs if t_calc < -t_α or t_calc > t_α.

Explanation: According to the study materials, for the test where σ1 and σ2 are unknown but assumed to be equal, the null hypothesis is H0: μ1 = μ2. The test statistic follows a t-distribution, not a Z-distribution. The degrees of freedom for this t-distribution are n1 + n2 - 2, not n1 + n2 - 1. For a two-tailed test, rejection of H0 occurs if t_calc < -t_α/2 or t_calc > t_α/2, not -t_α or t_α.

Question 3: The study concludes that employees working 8-hour days are absent for a shorter period of time than those working 10-hour days.

A. Yes

B. No

Explanation: The study materials provide the data for the two groups (10-hour days and 8-hour days) and state the objective: 'test to see whether the employees of the second group are absent for a shorter period of time than are the employees of the first group.' However, the materials do not present the conclusion or result of this hypothesis test, only the setup and the prompt to conduct the test.

Question 4: Based on the provided study material, what is the appropriate test statistic formula to use when investigating whether the mean sales prices generally differ between two car dealers, given sample means, sample standard deviations, and sample sizes for each dealer?

A. z_calc = (x̄ - μ0) / (σ / sqrt(n))

B. t_calc = (x̄1 - x̄2) / sqrt(σ1²/n1 + σ2²/n2)

C. z_calc = (p - π0) / sqrt(π0(1-π0)/n)

D. t_calc = ( (n1-1)s1² + (n2-1)s2² ) / (n1+n2-2) * (1/n1 + 1/n2) )

Explanation: The study material provides formula (g) under '3. Confidence intervals and statistics for hypothesis tests' as '2 2 2 1 2 1 2 1 ) ( n n x x z calc σ σ + − = or 2 2 2 1 2 1 2 1 ) ( n n x x t calc σ σ + − = '. This formula is used for comparing two means (x̄1 and x̄2) when standard deviations (σ1 and σ2) are given and sample sizes (n1 and n2) are used. Since the problem involves comparing mean sales prices between two dealers, option 1 directly represents the structure provided in formula (g) for comparing two means.

Question 5: The probability of an event A, denoted as P(A), can be greater than 1.

A. Yes

B. No

Explanation: The study materials state that the probability of an event P(A) is always within the range of 0 to 1, inclusive (0 ≤ P(A) ≤ 1). Therefore, it cannot be greater than 1.