In 1970, 59% of college freshmen thought that capital punishment should be abolished; by 2005, the percentage had dropped to 35%. Is the difference real, or can it be explained by chance? You may assume that the percentages are based on two independent simple random samples, each of size 1,000.

Assume that the two-dimensional vectors (X1, Y1), (X2, Y2), .......(Xn, Yn) form a random sample from a bivariate normal distribution for which the means of $X$ and $Y$ are unknown but the variances of $X$ and $Y$ and the correlation between $X$ and $Y$ are known. Find the M.L.E.s of the means.

Solve the equation using the Quadratic Formula.

$$x ^ { 2 } + 41 = - 8 x$$

A coin is tossed 500 times. Find the expected value and SE for the difference between the percentage of heads in the first 400 tosses and the percentage of heads in the last 100 tosses.

In a study of the stability of IQ scores, a large group of individuals is tested once at age 18 and again at age 35. The following results are obtained.

$$\text { age } 18 : \text { average score } \approx 100 , \quad \mathrm { SD } \approx 15$$

$$\text { age 35: average score } \approx 100 , \quad \text { SD } \approx 15 , \quad r \approx 0.80$$

a) Estimate the average score at age 35 for all the individuals who scored 115 at age 18. b) Predict the score at age 35 for an individual who scored 115 at age 18.

Find the p-value for the following large-sample z tests: a. A right-tailed test with observed z=1.15 b. A two-tailed test with observed z=-2.78 c. A left-tailed test with observed z=-1.81

The Gallup poll asks respondents how they would rate the honesty and ethical standards of people in different fields----very high, high, average, low, or very low. The percentage who rated clergy "very high or high" dropped fro,m 60% in 2000 to 54% in 2005. This may have been a chance variation. (You may assume that in each year, the results are based on independent simple random samples of 1,000 persons in each year.) a) Should you make a one-sample z-test or a two-sample z-test? Why? b) Formulate the null and alternative hypotheses in terms of a box model. Do you need one box or two? Why? How many tickets go into each box? How many draws? What do the tickets show? What do the null and alternative hypotheses say about the box(es)? c) Can the difference between 60% and 54% be explained as a chance variation? Or was it the scandals? Or something else?

Either evaluate the given improper integral or show that it diverges. $\int_{-\infty}^{\infty}\left(e^{x}+e^{-x}\right) d x$

Box A has an average of 100 and an SD of 10. Box B has an average of 50 and an SD of 18. Now 25 draws are made at random with replacement from box A, and independently 36 draws are made at random with replacement from box B. Find the expected value and standard error for the difference between the average of the draws from box A and the average of the draws from box B.

In a certain class, midterm scores average out to 60 with an SD of 15, as do scores on the final. The correlation between midterm scores and final scores is about 0.50. The scatter diagram is football-shaped. Predict the final score for a student whose midterm score is a) 75 b) 30 c) 60 d) unknown

A simple random sample of 3,500 people age 18 or over is taken in a large town to estimate the percentage of people (age 18 and over in that town) who read newspapers. It turns out that 2,487 people in the sample are newspaper readers. The population percentage is estimated as

$$\frac { 2,487 } { 3,500 } \times 100 \% \approx 71 \%$$

The standard error is estimated as 0.8 of 1%, because

$$\sqrt { 3,500 } \times \sqrt { 0.71 \times 0.29 } \approx 27 , \quad \frac { 27 } { 3,500 } \times 100 \% \approx 0.8 \text { of } 1 \%$$

a) Is 0.8 of 1% the right SE? Answer yes or no, and explain. b)

$$71 \% \pm 1.6 \%$$

is a _____ for the _____. Fill in the blanks and explain.

A geography test was given to a simple random sample of 250 high-school students in a certain large school district. One question involved an outline map of Europe, with the countries identified only by number. The students were asked to pick out Great Britain and France. As it turned out, 65.6% could find France, compared to 70.4% for Great Britain. Is the difference statistically significant? Or can this be determined from the information given?

Among freshman at a certain university, scores on the Math SAT followed the normal curve, with an average of 550 and an SD of 100. Fill in the blanks; explain briefly. a) A student who scored 400 on the Math SAT was at the ______ th percentile of the score distribution. b) To be at the 75th percentile of the distribution, a student needed a score of about ______ points on the Math SAT.

In 1990 and 2004, NAEP tested the 17-year-olds on mathematics as well as reading. The average score went up from 305 to 307. You may assume the NAEP took simple random samples of size 1,000 in each of the two years; the SD for the 1990 data was 34, and the SD for the 2004 data was 27. (In fact, NAEP used a more complicated sample design.) Can the difference between the 305 and 307 be explained as a chance variation? a) Should you make a one-sample z-test or a two-sample z-test? Why? b) Formulate the null and alternative hypotheses in terms of a box model. Do you need one box o two? Why? How many tickets go into each box? How many draws? Do the tickets show test scores, or 0's and 1's. Why? c) Now answer the main question: is the difference real, or can it be explained by chance/