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follow the steps below to construct a 90% confidence interval for the p…

Question

follow the steps below to construct a 90% confidence interval for the population proportion of all winning scratchers. (if necessary, consult a list of formulas.)
(a) click on \take sample\ to see the results from the random sample.
(b) based on your sample, enter the lower and upper limits to graph the 90% confidence interval for the population proportion of all winning scratchers.

Explanation:

Step1: Calculate the point estimate

The point estimate for the population proportion \(p\) is the sample proportion \(\hat{p}\). Given that the number of winning scratchers is \(x = 14\) and the sample size \(n=35\).

$$ \hat{p}=\frac{x}{n}=\frac{14}{35} = 0.4 $$

Step2: Calculate the standard error

The formula for the standard error of a proportion is \(SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). Substitute \(\hat{p}=0.4\) and \(n = 35\) into the formula.

$$ SE=\sqrt{\frac{0.4\times(1 - 0.4)}{35}}=\sqrt{\frac{0.4\times0.6}{35}}=\sqrt{\frac{0.24}{35}}\approx\sqrt{0.006857}\approx0.0828 $$

Step3: Determine the critical value

For a \(90\%\) confidence interval, the significance level \(\alpha=1 - 0.90=0.10\), and \(\alpha/2=0.05\). From the standard normal distribution table, the critical value \(z_{\alpha/2}=z_{0.05}\approx1.645\)

Step4: Calculate the margin of error

The formula for the margin of error \(E\) for a proportion is \(E = z_{\alpha/2}\times SE\). Substitute \(z_{\alpha/2}=1.645\) and \(SE\approx0.0828\) into the formula.

$$ E=1.645\times0.0828\approx0.1363 $$

Step5: Calculate the confidence interval

The confidence interval for the population proportion \(p\) is \(\hat{p}-E

$$ 0.4- 0.1363
$$ 0.2637

Answer:

  • Point estimate: \(0.4\)
  • Standard error: \(\approx0.0828\)
  • Critical value: \(1.645\)
  • Margin of error: \(\approx0.1363\)
  • \(90\%\) confidence interval: \((0.2637,0.5363)\)