110年 - 110 國立陽明交通大學_碩士班考試入學試題：統計學#121574

1. (10 points in total) The scores of English in a class are normally distributed. The distribution is divided into three portions, 5% of the students' scores are below 60 and 10% are above 80. Try to find the mean and the standard deviation for this course.

(a). (5 points) State the null and alternative hypothesis (H₀and H₁). What statistic should be constructed for testing the hypothesis and explain why (t or Z statistic?)

(b). (5 points) Do you reject H₀at the 5% level?

(a). (10 points) Do the data provide sufficient evidence to contradict the manufacturer's claim? (HINT: To be generous, let a range of 2% equal 4σ.)

(b). (5 points) Find a 90% confidence interval for σ²

(c). (5 points) What assumptions are made when you choose a test statistic for the hypothesis test above? Explain.

(a). (10 points) Is there sufficient evidence to indicate that our percentages of the colors for compact/sports cars differ from those given? Find the approximate p-value for the test (HINT: Remember to include a column called "Other" for cars that do not fall into one of the six categories shown in the table).

(b). (10 points) The researcher wants to see if there is a difference in the color distributions for compact/sports cars versus full/intermediate cars. Another random sample of 250 full/intermediate cars was taken and the color of the vehicles was recorded. The table below shows the results for both compact/sports and full/intermediate cars. Do the data indicate that there is a difference in the color distributions depending on the type of vehicle? Use α=0.05.

(a). (15 points) Please construct an ANOVA table to test whether the data present sufficient evidence to indicate that the number of errors is related to the number of hours without sleep? What is the p-value?

(b). (10 points) Please construct a simple linear regression to test whether the data present sufficient evidence to indicate that the number of errors is related to the number of hours without sleep. What is the least-squares prediction equation? y = a + βx + ε

(c). (10 points) What is the sampling distribution on β? (Hint: probability distribution, mean, and standard error)

(d). (5 points) How do you describe the strength of the relationship between y and x?

110年 - 110 國立陽明交通大學_碩士班考試入學試題：統計學#121574

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