aflacatn

2021-01-10

The Wall Street Journal reported that the age at first startup for $55\mathrm{%}$ of entrepreneurs was 29 years of age or less and the age at first startup for $45\mathrm{%}$ 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 $\stackrel{―}{p}$ where $\stackrel{―}{p}$ 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 $\stackrel{―}{p}$ where $\stackrel{―}{p}$ 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)?

Nathaniel Kramer

Given that the percentage of business owners whose first venture was launched when they were 29 years old or younger is $p=0.55$ and the sample size $n=200$
The proportion's sample distribution is roughly normal if
$np=200×0.55$
$=110⇒5$
$n\left(1—p\right)=200×\left(1—0.55\right)$
And
$=90⇒5$
As a result, the proportion's sample distribution is normal.
The mean of the  and standard deviation of
The percentage of business owners in this context whose initial startup occurred when they were 29 years old or younger is called bar p.
The mean of $\stackrel{―}{p}$ is
$E\stackrel{―}{p}\right)=p=0.55$
The standard deviation of $\stackrel{―}{p}$ is
${\sigma }_{p}=\frac{\sqrt{p\left(1-p\right)}}{n}=\frac{\sqrt{0.55×0.45}}{200}=0.0352$
So, the proportional sampling distribution $\stackrel{―}{p}$ an average number of entrepreneurs had their first business at age 29 or younger. $E\left(\stackrel{―}{p}\right)=0.55$ and standard deviation ${\sigma }_{p}=0.0352$

xleb123

Step a. We must figure out the standard error in order to display the sample distribution of $overlinep$. Standard error of a sample proportion is calculated as follows:
$S{E}_{\stackrel{―}{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$
Given that 55% of entrepreneurs had their first startup at 29 years of age or less, we can estimate $p$ as 0.55. Also, we have a sample size of $n=200$. Plugging these values into the formula, we get:
$S{E}_{{\stackrel{―}{p}}_{1}}=\sqrt{\frac{0.55\left(1-0.55\right)}{200}}$
Step b. Similarly, to show the sampling distribution of $\stackrel{―}{p}$ for entrepreneurs whose first startup was at 30 years of age or more, we can estimate $p$ as 0.45 (45%). Using the same sample size of $n=200$, the standard error becomes:
$S{E}_{{\stackrel{―}{p}}_{2}}=\sqrt{\frac{0.45\left(1-0.45\right)}{200}}$
Step c. To determine if the standard errors in parts (a) and (b) are different, we compare $S{E}_{{\stackrel{―}{p}}_{1}}$ and $S{E}_{{\stackrel{―}{p}}_{2}}$. If they are significantly different, it suggests that the sampling distributions of the two proportions are also different.

Jazz Frenia

a. To show the sampling distribution of $\stackrel{―}{p}$, where $\stackrel{―}{p}$ is the sample proportion of entrepreneurs whose first startup was at 29 years of age or less, we need to consider the binomial distribution.
Given that 55% of entrepreneurs had their first startup at 29 years of age or less, the probability of success for each entrepreneur is $p=0.55$. The sample size is $n=200$.
The mean of the sampling distribution is the same as the population proportion: ${\mu }_{\stackrel{―}{p}}=p=0.55$.
The standard error of the sampling distribution can be calculated using the formula:
${\sigma }_{\stackrel{―}{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$
Substituting the values, we have:
${\sigma }_{\stackrel{―}{p}}=\sqrt{\frac{0.55\left(1-0.55\right)}{200}}$
Therefore, the sampling distribution of $\stackrel{―}{p}$ for entrepreneurs whose first startup was at 29 years of age or less has a mean of 0.55 and a standard error of $\sqrt{\frac{0.55\left(1-0.55\right)}{200}}$.
b. To show the sampling distribution of $\stackrel{―}{p}$, where $\stackrel{―}{p}$ is the sample proportion of entrepreneurs whose first startup was at 30 years of age or more, we consider the complement of the previous case.
Given that 45% of entrepreneurs had their first startup at 30 years of age or more, the probability of success for each entrepreneur is $p=0.45$ (complement of the previous case). The sample size is still $n=200$.
The mean of the sampling distribution is the same as the population proportion: ${\mu }_{\stackrel{―}{p}}=p=0.45$.
The standard error of the sampling distribution can be calculated using the same formula:
${\sigma }_{\stackrel{―}{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$
Substituting the values, we have:
${\sigma }_{\stackrel{―}{p}}=\sqrt{\frac{0.45\left(1-0.45\right)}{200}}$
Therefore, the sampling distribution of $\stackrel{―}{p}$ for entrepreneurs whose first startup was at 30 years of age or more has a mean of 0.45 and a standard error of $\sqrt{\frac{0.45\left(1-0.45\right)}{200}}$.
c. No, the standard errors of the sampling distributions in parts (a) and (b) are not different. The standard error depends on the sample size and the probability of success (or failure) in the population, which in this case is either 0.55 or 0.45. Since the sample size and the probabilities are the same in both cases, the standard errors will also be the same.

fudzisako

Step 1:
a. The sample proportion, $\stackrel{―}{p}$, represents the proportion of entrepreneurs whose first startup was at 29 years of age or less. The sampling distribution of $\stackrel{―}{p}$ can be approximated by a normal distribution with mean ${\mu }_{\stackrel{―}{p}}$ and standard error ${\sigma }_{\stackrel{―}{p}}$.
Given that 55% of entrepreneurs have their first startup at 29 years of age or less, we can estimate ${\mu }_{\stackrel{―}{p}}$ as 0.55. Since we are sampling 200 entrepreneurs, the standard error can be calculated using the formula:
${\sigma }_{\stackrel{―}{p}}=\sqrt{\frac{{\mu }_{\stackrel{―}{p}}·\left(1-{\mu }_{\stackrel{―}{p}}\right)}{n}}$ where $n$ is the sample size.
Substituting the given values:
${\sigma }_{\stackrel{―}{p}}=\sqrt{\frac{0.55·\left(1-0.55\right)}{200}}$
Step 2:
b. In this case, $\stackrel{―}{p}$ represents the sample proportion of entrepreneurs whose first startup was at 30 years of age or more. The sampling distribution of $\stackrel{―}{p}$ can also be approximated by a normal distribution with mean ${\mu }_{\stackrel{―}{p}}$ and standard error ${\sigma }_{\stackrel{―}{p}}$.
Given that 45% of entrepreneurs have their first startup at 30 years of age or more, we can estimate ${\mu }_{\stackrel{―}{p}}$ as 0.45. The standard error can be calculated using the same formula as in part (a), since the sample size and the underlying population distribution are the same:
${\sigma }_{\stackrel{―}{p}}=\sqrt{\frac{0.45·\left(1-0.45\right)}{200}}$
Step 3:
c. The standard errors of the sampling distributions in parts (a) and (b) are the same. The standard error depends on the sample size and the underlying population distribution, which remain the same in both cases. Therefore, the standard errors for both sampling distributions are equal.

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