Answered: a. Construct the 95% confidence…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A random sample of 100 observations results in 70 successes. [You may find it useful to reference the z table.]

a. Construct the 95% confidence interval for the population proportion of successes. (Round intermediate calculations to at least 4 decimal places. Round "z" value and final answers to 3 decimal places.)

b. Construct the 95% confidence interval for the population proportion of failures. (Round intermediate calculations to at least 4 decimal places. Round "z" value and final answers to 3 decimal places.)

Expert Solution

Step 1

A random sample of 100 observations results in 70 successes. The population proportion of successes is,

$\begin{array}{rcl} \hat{p} & = & \frac{x}{n} \\ = & \frac{70}{100} \\ = & 0.70 \end{array}$

The proportion of successes is 0.70.

Computation of critical value:

$\begin{array}{rcl} (1 - α) & = & 0.95 \\ α & = & 0.05 \\ \frac{α}{2} & = & 0.025 \\ 1 - \frac{α}{2} & = & 0.975 \end{array}$

The critical value of z-distribution can be obtained using the excel formula “=NORM.S.INV(0.975)”. The critical value is 1.96.

The 95% confidence interval for the population proportion of successes is,

$\begin{array}{rcl} C I & = & \hat{p} \pm z_{\frac{α}{2}} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \\ = & 0.70 \pm 1.96 \sqrt{\frac{0.70 (1 - 0.70)}{100}} \\ = & 0.70 \pm 0.090 \\ = & (0.70 - 0.090, 0.70 + 0.090) \\ = & (0.610, 0.790) \end{array}$

Thus, the 95% confidence interval for the population proportion of successes is $\begin{array}{rcl} (0.610, 0.790) \end{array}$ .

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