Answered: A marketing researcher wants to estimate the mean amount spent per year ($) on a web site by membership member shoppers. Suppose a random sample of 100…

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Description: A marketing researcher wants to estimate the mean amount spent per year ($) on a web site by membership member shoppers. Suppose a random sample of 100 membership member shoppers who recently made a purchase on the web site yielded a mean amount spe

MATLAB: An Introduction with Applications

6th Edition

ISBN: 9781119256830

Author: Amos Gilat

Publisher: John Wiley & Sons Inc

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Question

A marketing researcher wants to estimate the mean amount spent per year ($) on a web site by membership member shoppers. Suppose a random sample of 100 membership member shoppers who recently made a purchase on the web site yielded a mean amount spent of $58
and a standard deviation of $55.
Complete parts (a) and (b) below.

a. Is there evidence that the population mean amount spent per year on the web site by membership member shoppers is different from

$51?

(Use a

0.01

level of significance.)

State the null and alternative hypotheses.

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

Hypothesis Set up and Test Statistic

Assuming the samples are drawn from a normal population.

$N u l l H y p o t h e s i s (H_{0}) : μ = $ 51 A l t e r n a t e H y p o t h e s i s (H_{1}) : μ \neq $ 51$

Given here is, $M e a n (μ) = 58, S \tan d a r d d e v i a t i o n (s) = 55, a n d S i z e o f s a m p l e (n) = 100$

$T h e t e s t s t a t i s t i c t i s, t = \frac{\bar{x} - μ}{\sqrt{\frac{S^{2}}{n}}} = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}} ~ t_{(n - 1)} w h e r e \frac{s}{\sqrt{n}} i s t h e s \tan d a r d a r d e r r o r .$

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