Problem 3

A new vegetable fertilizer is to be tested at two different levels (regular concentration and double concentration). Design an experiment, including a control, for 30 test plots, half of which are in shade. Explain carefully how you will use randomization.

Solution.

Problem 4

There are two types of lotteries, A and B. The price of the ticket of a lottery A is 150 Rub. With the probability 0.9 you may win 10 Rub and with probability 0.9 you may win 2 000 Rub. The price of the ticket of a lottery B is 150 Rub. With the probability 0.6 you win nothing and with probability 0.4 you may win 500 Rub. Your preference on a lottery is a tradeoff between the average net profit that you may win and the risk. Namely, suppose that your “utility” of a lottery is , where st.dev. is a standard deviation of a winning.

(a) What type of the lotteries, A or B, provides the highest expected net profit?

(b) What type of lottery, A or B, will you prefer?

Explain your answers.

Solution. Let’s calculate the expected values and standard deviations for both lotteries.

Lottery A: ,

and .

Lottery B: ,

and .

(a) Expected net profit for A = 209  150 = 59. Expected net profit for B = 200  150 = 50.

So, expected net profit in lottery A is greater than in lottery B.

(b) The utility of the lottery A = 59  0.05597 = 29.15.

The utility of the lottery B = 50  0.05244.95 = 37.72.

The utility of the lottery B is greater than the utility of the lottery A.

Problem 5

A pharmaceutical company claims that young children who are given additional vitamins and minerals in tablet form have an increased intelligence as measured on an IQ test.

In order to test this assertion two samples of children were selected. The first sample size was 220 and each child in this sample (the treatment sample) was given a daily tablet containing the additional vitamins and minerals; the second sample size was 420 and each child in this sample (the control sample) was given a daily tablet containing no active ingredients (a placebo).

After a certain period the children in the first sample were found to have an average increase in their IQ scores of 13 points and the corresponding average increase in the IQ’s of the second sample was 11 points. The respective sample variances were s₁² = 600 and s₂² = 520.

On the basis of these data, and using a 5% significance test, would you agree with the company’s claims?

Solution. Let be the mean increments of IQ for the first and the second groups. Then in order check the firm’s claim we should test the hypothesis H₀: versus H_a: . We have independent samples, and test-statistics is

where , and

. So . Under H₀ this statistics has tdistribution with 220 + 420  2= 638 degrees of freedom, that coincides with Zdistribution. One-sided Pvalue is 0.152 (25 points), and at 5% significance level we do not reject null hypothesis.

Section II Part b Problem 6

Researchers want to see whether training increase the capability of people to correctly predict the coin tosses. Each of twenty people is asked to predict the outcome (heads or tails) of 100 independent tosses of a fair coin. After training they are retested with a new set of 100 tosses. (all 40 sets of 100 tosses are independently generated.) Since the coin is fair, the probability of correct guess by chance is 0.5 on each toss. The numbers correct for each of the 20 people were as follows.

Score Before training (number correct)	46	48	50	54	54	54	54	54	54	54	55	56	57	58	58	61	61	63	64	65
Score After training (number correct)	61	62	53	46	50	52	53	59	60	61	55	59	55	50	56	58	64	57	61	54

To answer the following questions, you may want to enter these data into your calculator. As a check you have entered the data correctly, the sum of the first row is 1 120 and the sum of the second row is 1 126.

(a) Do the data suggest that after training people can correctly predict coin toss outcomes better than the 50 percent expected by chance guessing alone? Give appropriate statistical evidence to support your conclusion.

(b) Does the statistical test that you completed in part (a) provide evidence that this training is effective in improving a person’s ability to predict coin toss outcomes?

(c) Would knowing a person’s score before training be helpful in predicting his or her score after training? Justify your answer.

Solution.

(a) Let be the mean percent of correctly predicting outcomes after training. We should test the hypothesis H₀: versus H_a: . Test-statistics is which under H₀ has tdistribution with 19 degrees of freedom. We have . One-sided Pvalue is closed to 0, and the null hypothesis is rejected at any reasonable significance level. Thus we have strong statistical evidence that after training people can correctly predict the outcomes better than by chance.

(b), (c) No, in (a) we do not check the improving ability, but only compare with guessing by chance (10 points). Let be the mean percent of correctly predicting outcomes before training. In fact we have to test the hypothesis H₀: versus H_a: . Obviously we have matched sample. Test-statistics is which under H₀ has tdistribution with 19 degrees of freedom. We have . One-sided Pvalue is 0.42, and the null hypothesis is not rejected even at 20% significance level. So, we may conclude that the mean of correctly predicting outcomes does not change. It means that knowing the person’s score before training helps to predict his or her score after training: it would be likely the same.