107年 - 107 國立臺北大學_碩士班一般入學考試試題_統計學系：統計學#147169

1. (5%) Write down the probability distribution of X. You can specify some parameters.

2. (10%) Calculate the probability of more than 65 samples having \( X = 2 \) in terms of the parameters you specify in 1.

3. (5%) Figure I-A shows the distribution of the 100 sample observations of Y. Describe the center, the shape and the spread of this distribution.

4. (5%) Figure I-B shows the scatter plot of the sample observations of X and Y. The sample correlation of the two variables is -0.75. Do you think it is appropriate to use this measure to quantify the relationship between variables X and Y? Explain your reason.

5. (5%) Figure I-C shows the scatter plot of the sample observations of Y and Z. The text labels of the data point represent the corresponding observations of X. The sample correlation between Y and Z is -0.35. According to Figure I-C, do you think it is appropriate to use this measure to quantify the relationship between variables Y and Z? Explain your reason.

6. (10%) According to Figure I-C, write down an appropriate linear regression model to describe the relationship between variables Y and Z. Explain why you choose this model.

7. (5%) Interpret each parameter in your model.

8. (5%) According to your model, how would you determine whether there is a significant relationship between variables Y and Z?

(a) What is the probability that the sample mean service time is more than 65 minutes?

(b) The probability is 0.10 that the sample mean service time is less than how many minutes?

(c) The probability is 0.10 that the sample standard deviation of service times is less than how many minutes?

(d) What is the probability that more than two of these calls take more than 65 minutes?

(a) Assume it is known that $\sigma^2 = 4$, test $H0: \mu \ge 12$ versus $Ha: \mu < 12$ at the significance level $\alpha = 0.01$.

(b) What is the probability of a Type II error if in fact $\mu = 10.5$?

(c) What sample size is needed in order for the power of the test to be 0.90 for the alternative value $\mu = 10.5$?

(a) Does this data suggest that the true average ratio for young men exceeds that for elderly men? Carry out a test of appropriate hypotheses using $\alpha = 0.05$.

(b) Be sure to state any assumptions necessary for your analysis.

(c) In reaching your conclusion, what type of error might you have committed?