## Exercises forSampling Distributions & The Central Limit Theorem miscellaneous on-line topics for Finite Mathematics 2e

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1. Compute the sampling distribution for two tosses of a fair coin; $X = 1$ for heads, and $X = 0$ for tails.

2. Compute the sampling distribution for two tosses of a fair die; $X = 1$ for an even number, and $X = 0$ for an odd number.

3. Compute the sampling distribution for two tosses of an unfair coin, where $P(H) = 0.25$ and $P(T) = 0.75;$ $X = 1$ for heads, and $X = 0$ for tails.

4. Compute the sampling distribution for two tosses of a fair die, $X = 1$ for a $1$ or $2$ facing up, and $X = 0$ for any other number.

5. Consider the following probability ditribution.

 $X$ $0$ $1$ $5$ $P(X=x)$ $1/3$ $1/3$ $1/3$

(a) Find $μ.$     Answer
(b) Find the sampling distribution of the sample mean $\bar{x}$ for samples of size $n = 2.$     Answer
(c) Show that $X-$is an unbiased estimator for $μ.$     Answer

6. Consider the following probability ditribution.

 $X$ $-1$ $0$ $3$ $P(X=x)$ $1/4$ $1/4$ $1/2$

(a) Find $μ.$
(b) Find the sampling distribution of the sample mean $\bar{x}$ for samples of size $n = 2.$
(c) Show that $X-$is an unbiased estimator for $μ.$

7. Select the correct options to make the following sentence correct.

The Central Limit Theorem says that, if the population distribution has mean $μ.$ and standard deviation $σ,$ then, for samples of size $n,$ the sampling distribution of $\bar{x}$ is with mean $μ$ and standard deviation $σ_{\bar{x}} = σ/\sqrt{n}.$ For the theorem to hold, the population distribution .

8. Select the correct options to make the following sentence correct.

A statistic $S$ is an unbiased estimator of a population parameter $P$ of a random variable $X$ if the in its sampling distribution is the value of $P.$

9. According to a 1995 study, the mean family income in the US was $\$38,000$with a standard deviation of$21,000.$If a consulting agency surveys 49 families at random, what is the probability that it finds a mean family income of more than$\$41,500$?     Answer

10. According to coach Simon, the average height of male soccer players in the US is normally distributed with mean $1.79m$ with a standard devation of $0.04m.$
(a) In a randomly selected soccer team of 11 players, what is the probability that the averge height of the players is less than $1.77m$?
(b) In view of the small sample size, must you make any additional assumptions to justify the answer? Explain