Answered: A report published by the BZW Luxury…

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

A report published by the BZW Luxury Association shows that each American household spent on average
$694 on jewelry and watch in 2018. Suppose a sample of 64 households were surveyed while they were
shopping at Macy's, which resulted in a sample mean of $622 and a sample standard deviation of $125.

How does the margin of error for a 99% condence interval compare to the margin of error for a 95% condence interval?

Based on your the 95% confidence interval result, does it appear that the population mean amount spent by families shopping at Macy's is different from the mean reported? Explain.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

The confidence level is 99%.

For,

$\begin{array}{rcl} (1 - α) & = & 0.99 \\ α & = & 0.01 \end{array}$

Computation of critical value:

The critical value of t-distribution at 63 $(= 64 - 1)$ degrees of freedom can be obtained using the excel formula “=T.INV.2T(0.01,63)”. The critical value is 2.6561.

The 99% confidence interval for the population mean amount spent by families shopping at Macy's is,

$\begin{array}{rcl} C I & = & \bar{x} \pm M \\ = & \bar{x} \pm t_{\frac{α}{2}} (\frac{s}{\sqrt{n}}) \\ = & 622 \pm 2.6561 (\frac{125}{\sqrt{64}}) \\ = & 622 \pm (2.6561 \times 15.625) \\ = & 622 \pm 41.50 \\ = & (622 - 41.50, 622 + 41.50) \\ = & (580.5, 663.5) \end{array}$

Thus, the margin of error for 99% confidence interval is 41.50 and the 99% confidence interval for the population mean amount spent by families shopping at Macy's is $\begin{array}{rcl} (580.5, 663.5) \end{array}$ .

The confidence level is 95%.

For,

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

Computation of critical value:

The critical value of t-distribution at 63 $(= 64 - 1)$ degrees of freedom can be obtained using the excel formula “=T.INV.2T(0.05,63)”. The critical value is 1.9983.

The 95% confidence interval for the population mean amount spent by families shopping at Macy's is,

$\begin{array}{rcl} C I & = & \bar{x} \pm M \\ = & \bar{x} \pm t_{\frac{α}{2}} (\frac{s}{\sqrt{n}}) \\ = & 622 \pm 1.9983 (\frac{125}{\sqrt{64}}) \\ = & 622 \pm (1.9983 \times 15.625) \\ = & 622 \pm 31.22 \\ = & (622 - 31.22, 622 + 31.22) \\ = & (590.78, 653.22) \end{array}$

Thus, the margin of error for 95% confidence interval is 31.22 and the 95% confidence interval for the population mean amount spent by families shopping at Macy's is $\begin{array}{rcl} (590.78, 653.22) \end{array}$ .

It is clear that, the margin of error for 99% confidence interval is larger when compared to the margin of error for 95% confidence interval indicating that as confidence level decreases the margin of error also decreases.

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