112年 - 112 國立臺北大學_碩士班一般入學考試試題_統計學系：統計學#147118

科目：研究所、轉學考(插大)◆統計學 | 年份：112年 | 選擇題數：0 | 申論題數：17

試卷資訊

所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (0)

申論題 (17)

I. Researchers conducted a large-sample study to determine if smoking causes cancer. For each sample, they recorded information including smoking status (ever-smoker, never-smoker), weight, and cancer status (yes, no).

1. (10%) Use the researchers' data. Design a two-dimensional plot to display the association between weight and smoking status by cancer status. Explain your design.
2. (10%) Let \( X, W \), and \( Y \) be random variables representing smoking status, weight, and cancer status, respectively. Specify the three random variables by their sample spaces and distributions. Explain your notations and parameters.
3. (10%) Select appropriate measures.

(1) (5%) Write down a measure you would use to compare the weights between people of different cancer statuses.
(2) \((5\%)\) Write down a measure you would use to help the researchers answer their question.
4. (20%) Suppose the number of samples in the researchers' data in each category is summarized in the following table.

| | Ever Smoker | Never-Smoker |
| --- | --- | --- |
| Cancer Status Yes | 6020 | 5792 |
| Cancer Status No | 13983 | 24109 |

(1) \((5\%)\) Calculate an estimate of your proposed measure in 3. (2).
(2) \((10\%)\) Derive an approximate distribution of the estimator in 4. (1).
(3) \((5\%)\) Comment and explain: how well the distribution you derive in 4. (2) approximates the exact distribution.
II. The significance level is 0.05 for all questions in II unless otherwise specified.

1. (10%) A market research firm estimates the proportion of adults in a city who have tablets, and they believe more than half of the adults in this city have tablets. A confidence interval for the population proportion of adults have tablets from a random sample of 400 adults is constructed. The lower limit is 0.5597 and the upper limit is 0.6403.

(1) \((5\%)\) Determine the level of confidence used to construct the confidence interval.
(2) \((5\%)\) If another random sample of 400 adults is taken, and 218 adults have tablets. To test the hypothesis if more than half of the adults in the city have tablets, what is the P-value and your conclusion?
2. (10%) A doctor is studying whether walking leads to a reduction in systolic blood pressure (measured in mmHg). To answer the question, 9 volunteers began a one-week walking program. Their blood pressures were measured before and after the program and the data is as follows:

| Subject | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| before | 134 | 122 | 118 | 130 | 144 | 125 | 127 | 133 | 128 |
| after | 131 | 123 | 122 | 124 | 139 | 119 | 124 | 131 | 130 |

(1) \((5\%)\) Suppose the doctor would like to see if the data support that walking for one week will reduce the systolic blood pressure. What is the test statistics and your conclusion?
(2) \((5\%)\) Construct a \(95\%\) confidence interval for the reduction in systolic blood pressure.
3. (10%) The data of a class of students on the days of the week they are absent is as follows:

| Day | Mon | Tue | Wed | Thur | Fri |
| --- | --- | --- | --- | --- | --- |
| number of students absent | 13 | 9 | 6 | 8 | 14 |

The professor would like to know if the absence is equally likely to occur on any of the five days.

(1) (5%) Determine the test statistics for the hypothesis: the absence is equally likely to occur on any of the five days.
(2) (5%) What is the critical value for testing the hypothesis above? What is your conclusion?
4. (10%) We randomly selected 36 males and 25 females and compared the average time (in hours) they spent watching TV every day. The data are as follows:

| | mean | standard deviation |
| --- | --- | --- |
| males | 1.6 | 0.8 |
| females | 1.4 | 0.7 |

(1) (5%) To test if the average time spent watching TV is equal between male and female, what is the test statistics? Assume that the variance of time spent watching TV is equal for male and female.
(2) (5%) To test if the variance of time spent watching TV is equal for male and female, what is the test statistics and what is your conclusion?
5. (10%) A manager wants to test if different packaging (A, B, C, D) have an effect on coffee beans sale at supermarket. She randomly assigns each supermarket to one of the 4 packages and records packs of coffee beans sold in a one-week period.

| packaging | number of supermarkets | mean | variance |
| --- | --- | --- | --- |
| A | 9 | 48 | 23.50 |
| B | 9 | 50 | 39.00 |
| C | 8 | 56 | 20.86 |
| D | 8 | 59 | 44.57 |

The average coffee beans sold for all supermarkets in the week is 53 packs, and sum of square total (SST) is 1624.

(1) (5%) What is the sum of square between (among)?
(2) (5%) What is the test statistics for testing the hypothesis: there is no difference in sales of coffee beans for 4 different packaging? What is your conclusion?