# Obtain the sampling distribution of overline{p}

Question
Sampling distributions
Obtain the sampling distribution of $$\overline{p}$$

2021-02-10
It is given that is the sample proportion of entrepreneurs whose first startup was at 30 years or more is $$p = 0.45$$ and the sample size $$n = 200$$
The sampling distribution of the proportion is approximately normal if $$np \Rightarrow 5\ and\ n(l — p) \Rightarrow 5$$.
Verify the conditions:
$$np = 200 \times 0.45$$
$$=90\Rightarrow 5$$.
$$n(1 — p) = 200 \times (1 — 0.45)$$
And
$$=110\Rightarrow 5$$
The conditions are satisfied. Therefore, the sampling distribution of the proportion is normal.
The mean of the \overline{p} is $$E(\overline{p}) = p$$ and standard deviation of \overline{p} is $$\sigma_\overline{p}, = \frac{\sqrt{p(1-p)}}{n}$$
Mean and standard deviation are calculated as below
$$E(\overline{p}) =p = 0.45$$
$$\sigma_{\overline{p}}=\sqrt{p(1-p)}=\frac{\sqrt{0.45\times0.55}}{200}=0.0352$$
$$Py\ 0.45\times 0.55$$
Thus, the sampling distribution of the proportion J of entrepreneurs whose first startup was at 30 years or more is normal with mean $$E(\overline{p}) = 0.45$$ and standard deviation $$\sigma_{\overline{p}}, = 0.0352$$

### Relevant Questions

The Wall Street Journal reported that the age at first startup for $$55\%$$ of entrepreneurs was 29 years of age or less and the age at first startup for $$45\%$$ of entrepreneurs was 30 years of age or more.
a. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of \overline{p} where \overline{p} is the sample proportion of entrepreneurs whose first startup was at 29 years of age or less.
b. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of \overline{p} where \overline{p} is now the sample proportion of entrepreneurs whose first startup was at 30 years of age or more.
c. Are the standard errors of the sampling distributions different in parts (a) and (b)?
Critical Thinking Let x be a random variable representing the amount of sleep each adult in New York City got last night. Consider a sampling distribution of sample means $$\overline{x}$$.
How do the two $$\overline{x}$$ distributions for sample size $$n = 50\ and\ n = 100$$ compare?
Critical Thinking Let x be a random variable representing the amount of sleep each adult in New York City got last night. Consider a sampling distribution of sample means $$\overline{x}$$.
What value will the standard deviation $$\sigma_{\overline{x}}$$ of the sampling distribution approach?
Check whether the standard error of the sampling distributions of $$\overline{p}$$ obtained in part(a) and part(b) are different.
Suppose that $$\displaystyle{Y}_{{1}},{Y}_{{2}},{Y}_{{3}}$$ denote a random sample from anexponential distribution with density function $$\displaystyle{f{{\left({y}\right)}}}={\left({\frac{{{1}}}{{\lambda}}}\right)}\cdot{e}^{{{\frac{{-{y}}}{{\theta}}}}},{y}{>}{0}$$
=0 elsewhere
Consider the following five estimators of $$\displaystyle\theta$$:
$$\displaystyle\hat{{\theta}}_{{1}}={Y}_{{1}},\hat{{\theta}}_{{2}}={\frac{{{Y}_{{1}}+{Y}_{{2}}}}{{{2}}}},\hat{{\theta}}_{{3}}={\frac{{{Y}_{{1}}+{2}{Y}_{{2}}}}{{{3}}}},\hat{{\theta}}_{{4}}=\overline{{{Y}}}$$
a) Which of these estimators are unbiased?
b) Among the unbiased estimators, which has the smallest variance?
Which of the following statements about the sampling distribution of the sample mean is incorrect?
(a) The standard deviation of the sampling distribution will decrease as the sample size increases.
(b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples.
(c) The sample mean is an unbiased estimator of the population mean.
(d) The sampling distribution shows how the sample mean will vary in repeated samples.
(e) The sampling distribution shows how the sample was distributed around the sample mean.
The correct statement which is incorrect from the options about the sampling distribution of the sample mean
(a) the standard deviation of the sampling distribution will decrease as the sample size increases,
(b) the standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples,
(c) the sample mean is an unbiased estimator of the true population mean,
(d) the sampling distribution shows how the sample mean will vary in repeated samples,
(e) the sampling distributions shows how the sample was distributed around the sample mean.
The distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider two different random samples taken from the population, one of size 5 and one of size 85.
Which of the following is true about the sampling distributions of the sample mean for the two sample sizes?
Both distributions are approximately normal with mean 65 and standard deviation 3.5.
A
Both distributions are approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
B
Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.
C
Only the distribution for size 85 is approximately normal. Both distributions have mean 65 and standard deviation 3.5.
D
Only the distribution for size 85 is approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
E