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STAT111HWK4

STATISTICS 111

HOMEWORK 4 due 10/4/2013

NOTE: This Homework is longer than usual and carries more points than usual.

This homework is mainly about means and variances and calculations that can be made from

them. You will also sometimes have to use some formulas for the binomial distribution.

1(i). A fair coin is to be flipped 1,600 times. Use the two-standard-deviation rule to find twonumbers between which you are approximately 95% certain that the number of heads will

lie. [4 points]

1(ii). [Another window into statistics.] A coin (probability of head unknown) was flipped1,600 times and 851 of these flips give heads. What can you say, objectively, about whetheryou think it is a fair coin? [Any reasonable answer gets full points.] [3 points]

2. Consider the random variable T4, the sum of the numbers turning up on FOUR rolls ofa fair die. From your answer to Homework 3, Question 3, find the probability distributionof T4. (This will involve some tedious work, using however nothing more than commonsense. Think ofT4 as Y1 +Y2, where Y1 is the sum of the numbers turning up on rolls 1 and2 of the die, and Y2 is the sum of the numbers turning up on rolls 3 and 4 of the die. Nowthink (for example) of the ways in which T4 can equal 15. One way that this could happenwould be ifY1 = 3 and Y2 = 12. Another way that this could happen would be ifY1 = 4and Y2 = 11. And so on.) [10 points]

3. Use the long formulas for a mean and a variance to find the mean and variance ofT4 inQuestion 2 above. [9 points]

4. Use the mean and variance ofX1 (the random number to turn up on one roll of thedie), together with the relevant magic formulas (given in class) to find quickly the meanand variance ofT4 in Question 2 above. Do your answers agree with those that you found

in Question 3? Were they easier to calculate than they were in Question 3? [7 points]

5. Use the mean and variance ofX1 (the random number to turn up on one roll of a fair die),together with the relevant magic formulas (given in class) to find quickly the mean and vari-ance ofD (discussed in Question 7, Homework 3), and check that they agree with the valuesthat you found in Homework 3 by using the long formula for the mean and the long for-mula for the variance. [6 points]

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6. Use the mean and variance of X1, the random number to turn up on one roll of a

fair die, together with the relevant magic formulas (given in class) to find quickly themean and variance of the average of the four numbers to turn up on four rolls of thisdie. [5 points]

7. Suppose that the random variable X has a normal distribution with mean 0, variance 1.Find

(a)P(0 X 1.33), (b)P(0.37 X 0), (c)P(0.67 X 1.23). [6 points]

8. The random variable X has a normal distribution with mean 3, variance 4. Find

(a)P(2.05 X 3.33), (b)P(1.77 X 2.64), (c)P(3.78 X 4.19). [6 points]

9. Suppose that we know that the height of a woman chosen at random is a random variablehaving a normal distribution with mean 67, variance 9. (Height is measured in inches.) Find:-

(a) the probability that the height of a woman chosen at random will be between 65 and68, [4 points]

(b) the probability that the average of the heights of 20 women chosen at random will bebetween 65 and 68, [6 points]

(c) the probability that the average of the heights of 50 women chosen at random will bebetween 65 and 68. [6 points]

(d) Comment on the relationship between your answers to (a), (b) and (c). [3 points]

TOTAL: 75 points

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