Question

1. What is the Central Limit Theorem? Try to state it in your own words. 2. Consider the random variable x, where x is the number of dots after rolling a die. Make a sketch of the probability distribution of this variable. What is the expected value of x? 3. Now consider the random variable that is the average number of dots after four rolls. Is this variable normally distributed? Explain. 4. Suppose we changed the definition to the average number of dots after forty rolls. Would this variable be normally distributed? Explain. 5. Going forward, let’s define ?̅ as the average number of dots after forty rolls. a. What is the expected value of ?̅?   b. The standard deviation of x is 2.197. What is the standard error of ?̅?   6. What is the probability of obtaining an average that is less than 4.25? 7. What is the probability of obtaining an average that is within 0.5 of the expected value? Make a sketch to illustrate the probability. 8. Suppose that you increased the number of rolls to 100. You again calculate the probability that the average is within 0.5 of the expected value. Is this probability less than or greater than the probability you calculated in question 7? Why? (Try to answer without doing any calculations.) 9. What must be true so that the sampling distribution of ?̅ follows the normal distribution? 10. The probability of winning at the board game Monopoly is 32.5% if you move first. If you play 20 games of Monopoly where you move first, what is the probability that you win at least 10 out of 20 times? a. Express 10 out of 20 as a proportion. b. What is the population proportion? c. Calculate the standard error of the sample proportion. d. Now compute the probability of winning at least 10 out of 20 times.

Answer 1

1)

As the sample size increases, the sampling distribution of the mean follows the normal distribution such that for a population with a mean of mu and a standard deviation of sigma, the samples are drawn and if the sample is sufficiently large, the distribution of sample will follow a normal distribution.

2)

The random variable, x = number of dots after rolling a die.

The probability distribution of x is as follows,

x	P(x)
1	1/6
2	1/6
3	1/6
4	1/6
5	1/6
6	1/6

The expected value is obtained using the following formula,

3.

Since the sample size = 4 is too small, the central limit theorem can not be applied which requires the sample size of alt least 30. Hence the distribution of the average number of dots will not be normally distributed

4)

Now, the sample size = 40 which is greater than the required sample size of 30 to apply a central limit theorem. Hence the distribution of the average number of dots will be normally distributed

5)

a)

The expected value will be the same as calculated in part 2,

b)

The standard error is obtained using the following formula,

6)

The required probability is obtained by calculating the z score,

From the z distribution table,

7)

The required probability is obtained by calculating the z score,

From the z distribution table,

8)

Now, the standard error will be changed as shown below,

The required probability is obtained by calculating the z score,

From the z distribution table,

Conclusion: The probability is increased as the sample size increased. As the sample size increases, the standard error decreases which means the average approaches to the population mean.

9)

The sample size should be sufficiently large (i.e. n>30)

10)

a)

b)

c)

d)

The sample size of 20 games follows a binomial distribution with n = 20 and the probability of success = 0.325. SInce np = 20*0.325 > 5, we can take the normal approximation to the binomial.

The required probability is obtained by calculating the z score,

From the z distribution table,