114 年 國立中央大學企業管理學系碩士班乙組《統計學》

Which of the following statements concerning outlier detection is false?

(A) The box plot method uses the quartiles as a basis for detecting outliers.
(B) The $z$-score method is better at detecting outliers than the box plot method since it is less affected by the extreme observation in the data set.
(C) The $z$-score method uses the mean and standard deviation as a basis for detecting outliers.
(D) The box plot method is better at detecting outliers than the $z$-score method since it is less affected by the extreme observation in the data set.

Investing is a game of chance. Suppose there is a $30\%$ chance that a risky stock investment will end up in a total loss of your investment. Because the rewards are so high, you decide to invest in four independent risky stocks.

Find the probability that at least one of your investments becomes a total loss.

(A) $0.0081$
(B) $0.2401$
(C) $0.7599$
(D) $0.9919$

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of $1$.

Find the probability of observing exactly $3$ accidents on this stretch of road next two months. (Note: $e=2.7183$)

(A) $0.0613$
(B) $0.4905$
(C) $0.0226$
(D) $0.1804$

The amount of time it takes to take an exam has a skewed left distribution with a mean of $65$ minutes and a standard deviation of $4$ minutes.

If $64$ students were randomly sampled, find the probability that the sample mean of the sampled students exceeds $66$ minutes.

(A) $0.0228$
(B) $0.3085$
(C) $0.4772$
(D) Approximately $0$

Suppose a $95\%$ confidence interval for $p$ turns out to be $(0.36,0.44)$. Give a definition of what it means to be "$95\%$ confident" in an inference.

(A) In repeated sampling, the population parameter $p$ would fall in the given interval $95\%$ of the time.
(B) $95\%$ of the observations in the entire population fall in the given interval.
(C) In repeated sampling, $95\%$ of the intervals constructed would contain the population parameter $p$.
(D) $95\%$ of the observations in the sample fall in the given interval.

A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects $200$ students and finds that $118$ of them are receiving financial aid.

If the dean wanted to estimate the proportion of all students receiving financial aid to within $3\%$ with $95\%$ reliability, how many students would need to be sampled?

(A) $1068$
(B) $1033$
(C) $752$
(D) $728$

Suppose we wish to test $H_0:\mu=60$ vs. $H_1:\mu<60$. Which of the following possible sample results gives more evidence to suppose $H_1$ (i.e., reject $H_0$)?

(A) $\bar{x}=57,\ s=5$
(B) $\bar{x}=58,\ s=4$
(C) $\bar{x}=62,\ s=2$
(D) $\bar{x}=56,\ s=10$

Suppose the population is normal and the variance is unknown. If I test $H_0:\mu=70$ vs. $H_1:\mu\ne70$ and my test statistic is $-2.85$ for $n=16$, then my $p$-value for this test will be:

(A) $0.0044$
(B) $0.005<p\text{-value}<0.01$
(C) $0.01<p\text{-value}<0.02$
(D) $0.02<p\text{-value}<0.05$

The standard error of the mean for a sample of $100$ is $30$. In order to cut the standard error of the mean to $15$, we would

(A) increase the sample size to $200$.
(B) increase the sample size to $400$.
(C) decrease the sample size to $50$.
(D) decrease the sample size to $25$.

The probability distribution shown below describes a population of measurements.

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In a controlled laboratory environment, a random sample of $10$ adults and a random sample of $10$ children were tested by a psychologist to determine the room temperature that each person finds most comfortable. The data are summarized below:

🖼️【此處有附圖，請對照原卷】

Given $H_0:\mu=70$ vs. $H_1:\mu\ne70$, and $p\text{-value}=0.033$. What conclusion would you draw at the $0.01$ level of significance?

(A) Accept $H_0$
(B) Fail to reject $H_0$
(C) Reject $H_0$
(D) Not sufficient information to decide

The time between landings of airplanes at the Taoyuan airport follows an exponential distribution with a mean of $30$ seconds.

Find the probability that there will be between a $15$ second and a $45$ second gap between landings at the Taoyuan airport. (Note: $e=2.7183$)

(A) $0.6065$
(B) $0.2231$
(C) $0.8297$
(D) $0.3834$

Assume that $(X_1,X_2,X_3)$ is a random sample collected from a population with mean $\mu$ and variance $\sigma^2$.

Consider three point estimators of $\mu$:

$\hat{\mu}_1=X_2,\qquad \hat{\mu}_2=X_1-X_2+X_3,\qquad \hat{\mu}_3=\frac{X_1+2X_2+X_3}{4}.$

Which of the following statements is correct?

(A) $\hat{\mu}_1$, $\hat{\mu}_2$ and $\hat{\mu}_3$ are all unbiased.
(B) $\hat{\mu}_1$, $\hat{\mu}_2$ and $\hat{\mu}_3$ are all biased.
(C) $\hat{\mu}_1$ is more efficient.
(D) $\hat{\mu}_2$ is more efficient.

Which of the following is an appropriate null hypothesis?

(A) The sample mean is not larger than $10$.
(B) The sample mean is larger than $10$.
(C) The population mean is larger than $10$.
(D) The population mean is not larger than $10$.

For a given sample size $n$, if the level of significance $\alpha$ is increased, the power of the test

(A) will decrease.
(B) will increase.
(C) will remain the same.
(D) cannot be determined.

The test for the equality of two population variances is based on

(A) the difference between the two sample variances.
(B) the difference between the two population variances.
(C) the ratio of the two sample variances.
(D) the difference between the sample variances divided by the difference between the sample means.

Which of the following components in an ANOVA table are not additive?

(A) Sum of squares
(B) Degrees of freedom
(C) Mean squares
(D) It is not possible to tell.

The principal focus of the control chart is the attempt to separate special or assignable causes of variation from common causes of variation.

What cause of variation can be reduced only by changing the system?

(A) Common causes
(B) Special or assignable causes
(C) Total causes
(D) None of the above.

In performing a regression analysis involving two numerical variables, we are assuming

(A) the variances of $X$ and $Y$ are equal.
(B) that $X$ and $Y$ are independent.
(C) the expected value of $Y$ for each $X$ value is the same.
(D) the variation around the line of regression is the same for each $X$ value.

Let $X$, $Y$, and $Z$ be uncorrelated random variables with variances $\sigma_X^2$, $\sigma_Y^2$, and $\sigma_Z^2$, respectively. Let

$U=Z+X\qquad\text{and}\qquad V=Z+Y.$

(1) Find $\operatorname{Cov}(U,V)$. (4 pts)

(2) Find $\rho_{UV}$. (4 pts)

The average time a professional baseball player used to take to run from home base to first base was $3.79$ seconds. Last year, he underwent repair surgery due to a knee injury while baserunning. Although he has now fully recovered, the coach suspects that the professional baseball player's baserunning speed is not up to the past level and wants to conduct a hypothesis test.

(1) Please state the null hypothesis and alternative hypothesis. (3 pts)

(2) After setting the significance level to $0.05$, the coach decides to conduct $4$ baserunning tests from home base to first base. What test statistic should he choose? Please also describe the conditions of use and sampling distribution. (6 pts)

(3) Please describe the rejection region. (3 pts)

(4) The baserunning times are recorded as $3.87$, $3.78$, $3.91$ and $4.04$. What is the value of the test statistic? (3 pts)

(5) What is the conclusion about this test? (3 pts)

A model relating a person's wage ($wage$) to observed education ($educ$) and other unobserved factors ($u$) is specified as

$wage=\beta_0+\beta_1educ+u,$

where $wage$ is measured in dollars per hour and $educ$ is years of education.

(1) Please state the population regression function. (3 pts)

(2) The following table contains the years of education and wages per hour for $8$ laborers. Please estimate the relationship between $wage$ and $educ$ using OLS. (4 pts)

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