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ANEK [815]
3 years ago
7

Describe the sampling distribution of p(hat). Assume the size of the population is 30,000.

Mathematics
1 answer:
Naya [18.7K]3 years ago
7 0

Answer:

a) \mathbf{\mu_ \hat p = 0.6}

b) \mathbf{\sigma_p =0.01732}

Step-by-step explanation:

Given that:

population mean \mu = 30,000

sample size n = 800

population proportion p = 0.6

a)

The mean of the the sampling distribution is equal to the population proportion.

\mu_ \hat p =  p

\mathbf{\mu_ \hat p = 0.6}

b)

The standard deviation of the sampling distribution can be estimated by using the formula:

\sigma_p = \sqrt{\dfrac{p(1-p)}{n}}

\sigma_p = \sqrt{\dfrac{0.6(1-0.6)}{800}}

\sigma_p = \sqrt{\dfrac{0.6(0.4)}{800}}

\sigma_p = \sqrt{\dfrac{0.24}{800}}

\sigma_p = \sqrt{3 \times 10^{-4}}

\mathbf{\sigma_p =0.01732}

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As a rule of thumb, the sampling distribution of the sample proportion can be approximated by a normal probability distribution whenever the sample size is large.

<h3>What is the Central limit theorem?</h3>
  • The Central limit theorem says that the normal probability distribution is used to approximate the sampling distribution of the sample proportions and sample means whenever the sample size is large.
  • Approximation of the distribution occurs when the sample size is greater than or equal to 30 and n(1 - p) ≥ 5.

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