Answered: Use the sample information x¯x¯ = 43, σ…

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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Question

Use the sample information x¯x¯ = 43, σ = 6, n = 13 to calculate the following confidence intervals for μ assuming the sample is from a normal population.

(a) 90 percent confidence. (Round your answers to 4 decimal places.)

The 90% confidence interval is from __________ to _____________

(b) 95 percent confidence. (Round your answers to 4 decimal places.)

The 95% confidence interval is from____________ to_______________

(c) 99 percent confidence. (Round your answers to 4 decimal places.)

The 99% confidence interval is from_____________to________________

Expert Solution

Step 1

Given Information:

Sample mean $\bar{x} = 43$

Population standard deviation $σ = 6$

Sample size $n = 13$

Since, population standard deviation is known and the sample is drawn from a normal population, we use z-distribution.

(a) To construct 90% confidence interval:

Confidence level = 90% = 0.90

Significance level $α = 1 - 0.90 = 0.10$

Formula used to calculate confidence interval for $μ$ is:

$C . I = (\bar{x} - z_{α / 2} \frac{σ}{\sqrt{n}}, \bar{x} + z_{α / 2} \frac{σ}{\sqrt{n}})$

where, $z_{α / 2}$ is the critical value at $\frac{0.10}{2} = 0.05$ significance level.

Using standard normal table, z-critical value at 0.05 significance level is obtained as 1.645

Substitute the values in the formula:

$\begin{array}{rcl} C . I & = & (\bar{x} - z_{α / 2} \frac{σ}{\sqrt{n}}, \bar{x} + z_{α / 2} \frac{σ}{\sqrt{n}}) \\ = & (43 - 1.645 \frac{6}{\sqrt{13}}, 43 + 1.645 \frac{6}{\sqrt{13}}) \\ = & (43 - 1.645 \times 1.664100589, 43 + 1.645 \times 1.664100589) \\ = & (43 - 2.737445468, 43 + 2.737445468) \\ = & (40.26255453, 45.73744547) \\ \approx & (40.2626, 45.7374) \end{array}$

Thus, 90% confidence interval is from 40.2626 to 45.7374

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