Test on Statistics Exam: Time Series and Probability

Question 1: If the estimated least-squares trend line equation is $\hat{y} = 605.67 + 2.68x$ (with 2018 as the year of origin for the x-values), then the predicted number of cancer-related deaths in 2030 would be 640.6 (thousands).

A. Ano

B. Ne

Explanation: To predict the number of cancer-related deaths in 2030 using the given least-squares trend line $\hat{y} = 605.67 + 2.68x$ with 2018 as the year of origin, first calculate the x-value for 2030. The year 2018 corresponds to x=0, so 2030 corresponds to x = 2030 - 2018 = 12. Substitute x=12 into the equation: $\hat{y} = 605.67 + 2.68(12) = 605.67 + 32.16 = 637.83$. Therefore, the predicted number of deaths is 637.83 thousand, not 640.6 thousand.

Question 2: In the Z-test for comparing two proportions, given by the formula $Z = \frac{\hat{P}_1 - \hat{P}_2}{\sqrt{\hat{P}(1 - \hat{P})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$, what does the symbol $\hat{P}$ represent?

A. The sample proportion of the first group.

B. The sample proportion of the second group.

C. The pooled sample proportion.

D. The population proportion.

Explanation: The study materials state that $Z = \frac{\hat{P}_1 - \hat{P}_2}{\sqrt{\hat{P}(1 - \hat{P})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$, where $\hat{P} = \frac{X_1 + X_2}{n_1 + n_2}$. This $\hat{P}$ is the combined or pooled estimate of the common population proportion under the null hypothesis.

Question 3: The Central Limit Theorem states that the distribution of the sample mean is approximately normal only when the population distribution is itself normal.

A. Ano

B. Ne

Explanation: According to the Central Limit Theorem as stated in the study materials, 'Whatever the population distribution, the distribution of the sample mean is approximately normal when the sample is large.' This indicates that the theorem applies regardless of the population distribution, not only when it is normal.

Question 4: Which of the following statements about the normal distribution is incorrect?

A. In a normal distribution, most of the data values are clustered around the mean, and the distribution is symmetric around the mean.

B. The standard normal distribution has a bell-shaped density curve with a mean of zero and a variance of one.

C. For a normal distribution, a smaller standard deviation leads to a narrower and taller curve because the data are more tightly clustered around the variance.

D. If X follows a normal distribution with mean μ and variance σ², then the standardized variable Z = (X - μ)/σ follows the standard normal distribution.

Explanation: Statement C is incorrect. While a smaller standard deviation does result in a narrower and taller curve, it is because the data are more tightly clustered around the mean, not the variance. Variance is a measure of spread, but data clustering occurs around the central tendency (mean). Statements A, B, and D accurately describe properties of the normal distribution, as indicated in the study material.

Question 5: Irregular variation in a time series can be caused by natural events such as floods.

A. Ano

B. Ne

Explanation: Irregular variation (I) in a time series is caused by natural events such as floods or human events such as wars.