# Recent questions in College Statistics

Random variables

### Anystate Auto Insurance Company took a random sample of 370 insurance claims paid out during a 1-year period. The average claim paid was \$1570. Assume $$\sigma\ math=250$$. Find 0.90 and 0.99 confidence intervals for the mean claim payment.

Confidence intervals

### Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance $$\displaystyle\sigma^{{{2}}}$$ and (b) the population standard deviation $$\displaystyle\sigma$$. Interpret the results. The maximum wind speeds (in knots) of 13 randomly selected hurricanes that have hit the U.S. mainland are listed. Use a 95% level of confidence. $$\begin{matrix} 70 & 85 & 70 & 75 & 100 & 100 & 110 & 105 & 130 & 75 & 85 & 75 & 70 \end{matrix}$$

Confidence intervals

Random variables

Scatterplots

Study design

Two-way tables

### Describe how you can use a two -way table to organize data you collect from a survey.

Analyzing categorical data

Random variables

### A random variable X has the discrete uniform distribution $$f(x;k)=\frac{1}{k},x=1,2...,k$$ $$f(x;k)=0,$$ elsewhere. Show that the moment-generating function of X is $$\displaystyle{M}_{{{x}}}{\left({t}\right)}={\frac{{{e}^{{{t}}}{\left({1}-{e}^{{{k}{t}}}\right)}}}{{{k}{\left({1}-{e}^{{{t}}}\right)}}}}$$.

Confidence intervals

Random variables

### You randomly survey students about participating in the science fair. The two-way table shows the results. How many female students do not participate in the science fair? $$\begin{array}{|c|c|}\hline & \text{No} & \text{Yes} \\ \hline \text{Gender} \\ \hline \text{Female} & 15 & 22 \\ \hline \text{Male} & 12 & 32 \\ \hline \end{array}$$

Confidence intervals

### In the article "On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals," by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement: "Independent simple random samples, each of size 200, have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute." Based on the preceding results, what should you conclude about the equality of $$p_{1}$$ and $$p_{2}$$? Which of the three preceding methods is least effective in testing for the equality of $$p_{1}$$ and $$p_{2}$$?

Confidence intervals

Study design

### Do piano lessons improve the spatial-temporal reasoning of preschool children? A study designed to investigate this question measured the spatial-temporal reasoning of a random sample of 34 preschool children before and after 6 months of piano lessons. The differences (After - Before) in the reasoning scores have mean 3.618 and standard deviation 3.055.

Bivariate numerical data

Random variables

Two-way tables

### A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-fifties: \text{Soccer level} \ \begin{array}{ll|c|c|c} & & \text { Ellte } & \text { Non-elite } & \text { Did not play } \ \hline \text { Whether person } & \text { Yes } & 10 & 9 & 24 \ \hline \text { developed arthritis } & \text { No } & 61 & 206 & 548 \end{array} What percent of the elite soccer players developed arthritis? What percent of those who got arthritis were elite soccer players?

Significance tests

### Suppose that you want to perform a hypothesis test based on independent random samples to compare the means of two populations. You know that the two distributions of the variable under consideration have the same shape and may be normal. You take the two samples and find that the data for one of the samples contain outliers. Which procedure would you use? Explain your answer.

Confidence intervals