Question

Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of p where p is the sample proportion of entrepreneurs whose first startup was at 29 years of age or less. If required, round your answers to four decimal places.

np = 80
n(1-P) = 120
E = 0.4000
σP) = 0.0346

Answer 1

The question deals with the estimation of a population proportion using a normal approximation to the binomial distribution in statistics. Given that the sample size and the conditions (np > 5 and nq > 5) are met, the distribution can be modeled using a normal distribution with the given mean (sample proportion) and standard deviation.

Step-by-step explanation:

The student is asking a question related to the concept of sampling distributions and the estimation of a population proportion within the context of statistics. When dealing with sampling distributions of a proportion, the normal distribution can be used to approximate the binomial distribution given that certain conditions are met.

The conditions include having a sufficiently large sample size (commonly np > 5 and nq > 5 where q = 1 - p), and the trials are independent. Here, the student is given that the sample size is 200, the sample proportion of entrepreneurs (p') whose first startup was at 29 years or less is 0.2000 (np = 80), and the standard deviation of the sampling distribution (σP) is 0.0346.

For the given scenario, because np = 80 and nq = 120, both greater than 5, it is appropriate to use a normal distribution to model the sampling distribution of the sample proportion p. The mean of the distribution would be the sample proportion p' = 0.2, and the standard deviation would be σP = 0.0346, calculated as √npq/n, where n is the sample size.

Answer 2

The sampling distribution of the sample proportion p of entrepreneurs who started their business at age 29 or younger is modeled by a normal distribution with a mean of 0.4000 and a standard deviation of 0.0346, as both np and n(1-p) are greater than 5, fulfilling the requirements to approximate the binomial distribution with a normal one.

Step-by-step explanation:

The student's question relates to the sampling distribution of the sample proportion (p) of entrepreneurs starting a business at age 29 or younger. Given that np = 80 and n(1-p) = 120, where n represents the sample size and p indicates the proportion, we know that the conditions for the normal approximation to the binomial are satisfied, since both np and nq (which is n(1-p)) are greater than 5.

The sampling distribution of p can be approximated by a normal distribution with a mean (E) of 0.4000 (since np/n = 80/200) and a standard deviation (σP) of 0.0346. The normal distribution can be expressed as N(0.4, 0.03462). This allows us to make probability statements about the sample proportion using the normal distribution as an approximation.