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Description: Businesses know that customers often respond to background music. Do they also respond to odors? One study of this question took place in a small pizza restaurant in France on two Saturday evenings in May. On one of these evenings, a relaxing lavende

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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Businesses know that customers often respond to background music. Do they also respond to odors? One study of this question took place in a small pizza restaurant in France on two Saturday evenings in May. On one of these evenings, a relaxing lavender odor was spread through the restaurant. On the other evening no scent was used.

The data gives the time, in minutes, that two samples of 3030 customers spent in the restaurant and the amount they spent in euros.

No odor	Lavender
Minutes	Euros spent	Minutes	Euros spent
103103	15.915.9	9292	21.921.9
6868	18.518.5	126126	18.518.5
7979	15.915.9	114114	22.322.3
106106	18.518.5	106106	21.921.9
7272	18.518.5	8989	18.518.5
121121	21.921.9	137137	24.924.9
9292	15.915.9	9393	18.518.5
8484	15.915.9	7676	22.522.5
7272	15.915.9	9898	21.521.5
9292	15.915.9	108108	21.921.9
8585	15.915.9	124124	21.521.5
6969	18.518.5	105105	18.518.5
7373	18.518.5	129129	25.525.5
8787	18.518.5	103103	18.518.5
109109	20.520.5	107107	18.518.5
115115	18.518.5	109109	21.921.9
9191	18.518.5	9494	18.518.5
8484	15.915.9	105105	18.518.5
7676	15.915.9	102102	24.924.9
9696	15.915.9	108108	21.921.9
107107	18.518.5	9595	25.925.9
9898	18.518.5	121121	21.921.9
9292	15.915.9	109109	18.518.5
107107	18.518.5	104104	18.518.5
9393	15.915.9	116116	22.822.8
118118	18.518.5	8888	18.518.5
8787	15.915.9	109109	21.921.9
101101	25.525.5	9797	20.720.7
7575	12.912.9	101101	21.921.9
8686	15.915.9	106106	22.522.5

To access the complete data set, click the link for your preferred software format:

Excel Minitab JMP SPSS TI R Mac-TXT PC-TXT CSV CrunchIt!

The two evenings were comparable in many ways (weather, customer count, and so on), so we are willing to regard the data as independent SRSs from spring Saturday evenings at this restaurant. The authors say, "Therefore, at this stage it would be impossible to generalize the results to other restaurants."

(a) Does a lavender odor encourage customers to stay longer in the restaurant? Examine the time data and explain why they are suitable for two‑sample ?t procedures.

Because the distributions are reasonably symmetric with no outliers and the samples can be treated as independent SRSs, the ?t procedures will work well.

true

false

Let ?1μ1 be the population mean time in the restaurant with no scent, and ?2μ2 be the mean time with a lavender odor. State the null and alternative hypotheses.

?0:?1>?2H0:μ1>μ2 vs. ??:?1<?2Ha:μ1<μ2

?0:?1=?2H0:μ1=μ2 vs. ??:?1>?2Ha:μ1>μ2

?0:?1=?2H0:μ1=μ2 vs. ??:?1<?2Ha:μ1<μ2

?0:?1=?2H0:μ1=μ2 vs. ??:?1≠?2Ha:μ1≠μ2

Give the value of the test statistic. (Enter your answer rounded to two decimal places.)

test statistic:

Use the conservative degrees of freedom to give the P‑value for the test.

0.0005<?<0.0010.0005<P<0.001

?<0.0005P<0.0005

0.0025<?<0.0050.0025<P<0.005

0.01<?<0.020.01<P<0.02

What is your conclusion?

We have some evidence that the mean times spent when lavender odor is present and when there is no odor are not the same.

There is no evidence that the mean time spent in the restaurant when lavender odor is present is different from the mean time spent when there is no odor.

We have strong evidence that customers stay longer when the lavender odor is present.

We have sufficient evidence that customers stay longer when there is no odor.

(b) Does a lavender odor encourage customers to spend more while in the restaurant? Examine the spending data. In what ways do these data deviate from normality? Select your choices.

The distributions do not have similar shapes.

There are two clear asymmetrical peaks in each distribution.

The distributions are skewed and have many gaps.

There are extreme outliers.

If ?1μ1 is the population mean spending with no scent, and ?2μ2 is the mean spending with a lavender odor, what are the null and alternative hypotheses?

?0:?1=?2H0:μ1=μ2 versus ??:?1≠?2Ha:μ1≠μ2

?0:?1−?2=0H0:μ1−μ2=0 versus ??:?1−?2>0Ha:μ1−μ2>0

?0:?1=?2H0:μ1=μ2 versus ??:?1>?2Ha:μ1>μ2

?0:?1=?2H0:μ1=μ2 versus ??:?1<?2

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 + ...

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