Q65E Suppose you wish to estimate the... [FREE SOLUTION]

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Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset 91影视 Explanations$ Math

13th Edition 路 888 pages

ISBN: 9780134506593

Chapter 6: Q65E (page 363)

Suppose you wish to estimate the mean of a normal population
using a 95% confidence interval, and you know from prior information that $蟽^{2} 鈮� 1$
a. To see the effect of the sample size on the width of the confidence interval, calculate the width of the confidence interval for n= 16, 25, 49, 100, and 400.
b. Plot the width as a function of sample size non graph paper. Connect the points by a smooth curve and note how the width decreases as nincreases.

Short Answer

Expert verified

a. The width of the confidence interval for n=16 is 0.98, The width of the confidence interval for n=25 is 0.784, The width of the confidence interval for n=49 is 0.56, The width of the confidence interval for n=100 is 0.392 The width of the confidence interval for n=400 is 0.196

b.From the drawn graph, It is observed that the width of the confidence interval decrease as the sample size increase.

Step by step solution

Given information

Know from prior information that $蟽^{2} 鈮� 1$

Finding the width of the confidence interval

The width of the confidence interval is

$Width = 2 脳 z_{伪 / 2} 脳 \frac{蟽}{\sqrt{n}}$

The critical value for a 95% confidence interval is $z_{伪 / 2} = z_{0.025} = 1.96$

For sample size n=16

$\begin{matrix} Width = 2 脳 z_{伪 / 2} 脳 \frac{蟽}{\sqrt{n}} \\ = 2 脳 1.96 脳 \frac{1}{\sqrt{16}} \\ = 0.98 \end{matrix}$

For sample size n=25

$\begin{matrix} Width = 2 脳 z_{伪 / 2} 脳 \frac{蟽}{\sqrt{n}} \\ = 2 脳 1.96 脳 \frac{1}{\sqrt{25}} \\ = 0.784 \end{matrix}$

For sample size n=49

$\begin{matrix} Width = 2 脳 z_{伪 / 2} 脳 \frac{蟽}{\sqrt{n}} \\ = 2 脳 1.96 脳 \frac{1}{\sqrt{49}} \\ = 0.56 \end{matrix}$

For sample size n=100

$\begin{matrix} Width = 2 脳 z_{伪 / 2} 脳 \frac{蟽}{\sqrt{n}} \\ = 2 脳 1.96 脳 \frac{1}{\sqrt{100}} \\ = 0.392 \end{matrix}$

For sample size n=400

$\begin{matrix} Width = 2 脳 z_{伪 / 2} 脳 \frac{蟽}{\sqrt{n}} \\ = 2 脳 1.96 脳 \frac{1}{\sqrt{400}} \\ = 0.196 \end{matrix}$

Plotting the width of the confidence interval

Here it is observed that the width of confidence interval decrease as the sample size increase.

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91影视

Short Answer

Step by step solution

Given information

Finding the width of the confidence interval

Plotting the width of the confidence interval

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