**1.** To approximate the integral

, use R code.

**2. **To approximate the integral

, use R code.

**3. ** Use R code to build iris flower data set and generate the figure as bellow:

data(iris)

**4. **The following 16 numbers came from normal random number generator on a computer:

3.5281 4.7433 0.1077 1.5977 5.4920 1.7220 4.1547 2.2799

3.5281 4.7433 0.1077 1.5977 5.4920 1.7220 4.1547 2.2799

What would you guess the mean and variance (𝜇 and 𝜎 ^2 ) of the generating normal distribution were?

Give 90%, 95%, and 99% confidence intervals for 𝜇 and 𝜎^ 2 .

Give 90%, 95%, and 99% confidence intervals for 𝜎.

How much larger a sample do you think you would need to halve the length of the confidence interval for 𝜇?

5. Read the paper "Diagnosis of multiple cancer types by shrunken centroids of gene expression" by Tibshirani et al. (2001). Summarize your understanding. Compose with 200 to 300 words for this section.

5. Read the paper "Choosing between logistic regression and discriminant analysis?" by Press and Wilson (1978). Summarize your understanding and compose with 200 to 300 words for this section.

6. Describe what are overfitting and underfitting in Machine Learning? How to prevent them? Compose with 100 to 200 words for this section.

7. Use R code to build a perfect random number generators, and verify the perfection. There is a maximum of 200 words for this section, includes code and explanation.

8. True or false?

The significance level of a statistical test is equal to the probability that the null hypothesis is true.

If the significance level of a test is decreased, the power would be expected to increase.

If a test is rejected at the significance level α, the probability that the null hypothesis is true equals α.

The probability that the null hypothesis is falsely rejected is equal to the power of the test.

A type I error occurs when the test statistic falls in the rejection region of the test.

A type II error is more serious than a type I error.

The power of a test is determined by the null distribution of the test statistic.

The likelihood ratio is a random variable

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