109年 - 109 國立臺北大學_碩士班一般入學考試試題_統計學系：統計學#147129

一、(50%)

1. (10%) Identical twins (同卵雙胞胎) form when a single fertilized egg splits into two genetically identical parts. The identical twins share the same DNA set and are thus always of the same sex. Fraternal (異卵雙胞胎) twins develop when two separate eggs are fertilized and implant in the uterus. The genetic connection is the same as siblings (兄弟姐妹) born at separate times. If 64 percent of all twin pairs are of the same sex, what percentage of twin pairs are identical twins?

a. Find the sampling distribution when a pair of fair dice is rolled (mean and standard deviation)

b. Determine the variance of the sum obtained when a pair of fair dice is rolled.

3. (10%) The mean of a Poisson random variable X is μ = 4 with standard deviation σ.

Find P(μ - σ < X < μ + σ).

a. Find \( \mathrm{P}(12\leq X\leq 14) \)

b. Find the approximate value of \( P(12 \leq X \leq 14) \) using the normal approximation.

a. two population standard deviations can be assumed equal.

b. two population standard deviations can not be assumed equal.

a. State the ANOVA model for the experimental design (i.e. full model of the two-factor analysis of variance).

b. Fit the ANOVA model (i.e. estimate the model parameters).

c. Set up the ANOVA table and test for the significances of factor A, factor B, and interaction of factors A and B at the \( 5\% \) level of significance.

a. Construct a \( 90\% \) confidence interval for the true proportion of people who liked the chocolate bar.

b. With a 0.90 probability, how large of a sample needs to be taken to provide a margin of error of \( 3\% \) or less?

a. State the null and alternative hypotheses. Test the hypothesis at the \( 5\% \) level of significance. What do you conclude?

b. Compute the power of the statistical test when \( \mu = 40,000 \)

a. Develop the least squares estimated regression equation.

b. At \( 95\% \) confidence, perform a \( t \) test and determine whether or not the slope is significantly different from zero.

c. Perform an \( F \) test to determine whether or not the model is significant. Let \( \alpha = 0.05 \).

d. Compute the coefficient of determination. Explain it.

e. Construct a \( 95\% \) confidence interval and prediction interval of \( y \) with \( x = 5 \)