107年 - 107 國立臺北大學_碩士班一般入學考試試題_國際企業研究所：統計學#147489

I. (50%)

1.1.(10%) Let X have a distribution with p.d.f.

$$f(x) = \frac{e^{-x}}{(1+e^{-x})^2}, -\infty < x < \infty.$$

What is the distribution of $Y = 1/(1+e^{-X})$?

(a) What is the null hypothesis for this question?

(b) Write down the probability of making a Type II error when the population mean idle time is 80 minutes at .05 level of significance in a probability form without checking the Standard Normal table. \( (z{0.05} = 1.96 \) and \( z{0.05} = 1.645) \)

(a) What is the variance of \( \bar{D} \)?

(b) If the two-sample t test is incorrectly used to analyze paired data, what will happen if \( \rho > 0 \)?

1.4.(10%) Consider a random sample $X_i, i=1, \dots, n$. Let $\mu$ and $\sigma^2$ denote the

population mean and variance. Define $\bar{X} = n^{-1} \sum{i=1}^n Xi$, $s1^2 = n^{-1} \sum{i=1}^n (Xi - \mu)^2$, $s2^2 = n^{-1} \sum{i=1}^n (Xi - \bar{X})^2$, What are the distributions of $ns1^2/\sigma^2$ and $ns2^2/\sigma^2$ if n=50? (Justify your answers)

1.5.(10%) The following table shows part of the probability distribution for a random Variable \( x \). The mean of the distribution is known to be 1.8 (\( E(x) = 1.8 \)). Determine \( f(1) \) and \( f(3) \).

(a) State the null and alternative hypotheses to be tested.

(b) Compute the test statistic.

(c) The null hypothesis is to be tested at the \( 5\% \) level. Determine the critical value for this test. What do you conclude?

(d) Determine the \( p \)-value and perform the test.

(a) Fill in all the blanks in the above ANOVA table.

(b) At \( a = .05 \), test to see if there is a significant difference among the average heating bills of all homes in the city using 3 forms of heating. (State the null and alternative hypotheses, compute the test statistic, and make your conclusion).

(a) Fill in all the blanks in the following ANOVA table.

(b) Is the model significant? Let \( \alpha = .05 \). (State the null and alternative hypotheses, compute the test statistic, and make your conclusion).

(a) State the null and alternative hypotheses.

(b) At a .05 level of significance, test to determine if the two added independent variables contribute significantly to the model. (Compute the test statistic, and make your conclusion).