課程名稱︰統計導論
課程性質︰數學系選修
課程教師︰鄭明燕
開課系所︰數學系
考試時間︰January 12, 2006 15:30~17:30
試題 : // 紅色的部分表示配分, 總分130
// 下標由於打不出來, 所以我都用在前面加個底線表示
例如X_1, 表示 X 下標為 1 ; H_0 表示 H 下標為 0 , 即虛無假設
1. Confidence Interval.
(a) (5) Suppose, based on a sample, one has a level-95% confidence
interval (0,21, 0.35) for the probability of rain tomorrow.
What does 95% mean here?
(b) (5) Suppose one has 20 level-95% confidence intervals, based on 20
samples, for a population mean. How many of them will cover the
mean?
2. Explain the terms in hypothesis testing.
(a) (4) Type Ⅰ error and Type Ⅱ error.
(b) (6) level and power.
3. To test H_0:μ＝μ_0 versus H_1:μ≠μ_0 using an i.i.d. sample from
Normal(μ, σ^2), where σ^2 is unknown, give the three methods.
(a) (5) The traditional method.
(b) (5) The p-value method.
(c) (5) The confidence interval method.
4. Two means.
(a) (8) Define the two-sided level-(1－α) t-test for the difference of
two population means based on two independent samples.
(b) (8) Define the two-sided level-(1－α) t-test for the difference of
two population means based on a matched-pair sample.
(c) (4) What is purpose of matching pairs?
(d) (10) Define the p-values of the tests in (a) and (b).
(e) (5) Suppose the sample variances and samples sizes in (a) and (b)
are the same. Which method gives shorter confidence interval?
5. Correlation coefficient.
(a) (5) Define the correlation coefficient, denoted as ρ, of two random
variables Ｘ and Ｙ.
(b) (10) Give an example in which two variables are related but have
zero corrlation coefficient.
6. Chi-square tests.
(a) (6) Give two examples where one can apply the univariate chi-square
test.
(b) (7) Give the chi-square test of independence of two categorical
variables.
(c) (7) Give the chi-square test of homogeneity.
7. Linear regression.
(a) (5) Write down the linear regression model with one independent
variable. Give the model assumptions.
(b) (5) Define the least squares problem for fitting the regression line
to a sample (X_1, Y_1), ... , (X_n, Y_n).
(c) (5) Give the prediction interval for the response of a future point
(X_0, Y_0), which is independent of the sample (X_1, Y_1), ... ,
(X_n, Y_n) and obeys the model in (a).
8. ANOVA. Consider the one-way analysis of variance.
(a) (5) Write down the model and model assumptions.
(b) (5) Give the analysis of variance table.
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