1. For each Claim below, write: (1) claim in mathematical form; (2) Null hypothesis (Ho); (3) alternative hypothesis (H,); (4) what tailed test. Use notations Hi. Hz (for mean), P-Pz (for proportion), .02 (Sor standard deviation), Fo (for differences in means for dependent (Le, matched pair) samples) (10

(a) Claim: Califormia HS students on an average score lower in the SAT than Massachusetts HS students (1-Califomia, 2- Massachusets)

(b) Claim: The proportion of smokers in Texas is the same as that in Oklahoma (1- Texas, 2- Oklahoma)

(c) Claim: The standard deviation of weights in 8 oz Coke cans is at least as high as that in Pepsi cans (1- Coke, 2- Pepsi)

(d) Claim: Ethiopian track athletes are, on an average, faster long-distance runners than the USA track athletes (1 Ethiopia, 2- USA)

(e) Claim: The cholesterol level before taking Lipitor is, on an average, higher than that after taking the drug (1-Before drug. 2-Afer drug).

2. Average Temperatures The average temperatures for a 25-day period for Birmingham, Alabama, and Chicago, Illinois, are shown. Based on the samples, at a =0.10, can it be concluded that it is warmer in Birmingham?

Birmingham 78 82 68 67 68 75 73 75 64 68 62 73 77 78 79 74 72 73 78 68 73 79 82 71 66.

Chicage 70 74 73 60 77 71 72 71 74 76 71 80 65 70 83 67 76 7s 62 65 66 65 77 66 64.

3 Gambling A survey of 60 men found that 36 gamble. Another survey of 50 women found that 28 gamble. At a-0.01, is there a difference in the proportions?

4. Testing After Review A statistics class was given a pretest on probability (since many had previous experience in some other class). Then the class was given a six-page review handout to study for 2 days. At the next class they were given another test. Is there sufficient evidence that the scores improved? Use a-0.05.

students 1 2 3 4 5 6

pretest 52 50 40 58 60 52

Past test 62 65 50 65 68 63

5. For the problem below: a. Draw the scatter plot. b. Compute the value of the correlation coefficient. c. Test the significance of the correlation coefficient at a-0.01, using the shortcut method that you learned from the PowerPoint slides d. Determine the regression line equation if r is significant. e. Plot the regression line on the scatter plot, if appropriate. f. Predict y' for a specific value of x, if appropriate. Protein and Diastolic Blood Pressure A study was conducted with vegetarians to see whether the number of grams of protein each ate per day was related to diastolic blood pressure. The data are given here. If there is a significant relationship, predict the diastolic pressure of a vegetarian who consumes 8 grams of protein per day.

Grams x 4 6.5 5 5.5 8 10 9 8.2 10.5

pressure y 73 79 83 82 84 92 88 86 95

6.The world happiness report 2017 provides a happiness score for countries using the data collected by the Gallup World Poll. Assuming happiness score is approximately normal with mean 5.35 with a standard deviation of 1.13, answer the below questions.

What percentage of countries has a happiness score of at least 7.0?

What percentage of countries has a happiness score less than 3.0?

What percentage of countries has a happiness score in between 3.0 and 6.0?

7. please solve and explain:

Let X1 and X2 be independent random variables having the standard normal distribution. Obtain the joint Lebesgue density of (Y1, Yz), where Yi = VX?+ X} and Y2 = X1/X2. Are Y1 and Y2 independent? Note. For this type of problem, we may apply the following result. Let X be a random k-vector with a Lebesgue density fx and let Y = g(X), where g is a Borel function from (R*, B* ) to (R*, B*). Let A1, . .., Am, be disjoint sets in B* such that R* - (Aj U . . . U Am,) has Lebesgue measure 0 and g on A; is one-to-one with a nonvanishing Jacobian, i.e., the determinant Det(Og(x)/Or) # 0 on Aj, j =1. .... m. Then Y has the following Lebesgue density: fy (x) = >Det (Oh, (x)/ar) | fx (h,(r)). j=1 where h, is the inverse function of g on A;, j = 1, ..., m.Let X and Y be independent random variables with cumulative distribution functions Fx and Fy, respectively. Show that (i) the cumulative distribution function of X +Y is Fxty (t ) = Fy(t - r)dFx (r); (ii) Fxty is continuous if one of Fx and Fy is continuous; (iii) X + Y has a Lebesgue density if one of X and Y has a Lebesgue density