Sampling Distribution Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
A population proportion is p=0.40p=0.40. For samples of size n=100n=100, describe the sampling distribution of p^\hat{p} and find P(p^>0.45)P(\hat{p} > 0.45).

Solution

  1. 1
    Mean: ฮผp^=p=0.40\mu_{\hat{p}} = p = 0.40
  2. 2
    Standard error: SE=p(1โˆ’p)n=0.40ร—0.60100=0.0024โ‰ˆ0.049SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.40 \times 0.60}{100}} = \sqrt{0.0024} \approx 0.049
  3. 3
    Shape: approximately Normal by CLT (np=40โ‰ฅ10, n(1-p)=60โ‰ฅ10)
  4. 4
    P(p^>0.45)=P(Z>0.45โˆ’0.400.049)=P(Z>1.02)โ‰ˆ0.154P(\hat{p} > 0.45) = P\left(Z > \frac{0.45-0.40}{0.049}\right) = P(Z > 1.02) \approx 0.154

Answer

p^โˆผN(0.40,0.049)\hat{p} \sim N(0.40, 0.049); P(p^>0.45)โ‰ˆ0.154P(\hat{p} > 0.45) \approx 0.154.
The sampling distribution of p^\hat{p} parallels that of Xห‰\bar{X}: normally distributed by CLT with mean pp and standard error p(1โˆ’p)/n\sqrt{p(1-p)/n}. This is the foundation for proportion-based confidence intervals and hypothesis tests.

About Sampling Distribution

The probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

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More Sampling Distribution Examples

Example 1 medium

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Example 2 hard

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Example 3 easy

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