• paraphrase 1 page paper

The major shopping areas in the community of Springdale include Springdale Mall, West Mall,

and the downtown area on Main Street. A telephone survey has been conducted to identify

strengths and weaknesses of these areas and to find out how they fit into the shopping activities

of local residents. The 150 respondents were also asked to provide information about themselves

and their shopping habits. The data are provided in the file Shopping (attached below). The

variables in the survey can be found in the file Coding (attached below).

In this exercise, some of the estimation techniques presented in the module will be applied to the

Springfield Shopping survey results. You may assume that these respondents represent a simple

random sample of all potential respondents within the community, and that the population is

large enough that application of the finite population correction would not make an appreciable

difference in the results.

Managers associated with shopping areas like these find it useful to have point estimates

regarding variables describing the characteristics and behaviors of their customers. In addition, it

is helpful for them to have some idea as to the likely accuracy of these estimates. Therein lies the

benefit of the techniques presented in this module and applied here.

1. Item C in the description of the data collection instrument lists variables 7, 8, and 9, which
represent the respondent’s general attitude toward each of the three shopping areas. Each of
these variables has numerically equal distances between the possible responses, and for
purposes of analysis they may be considered to be of the interval scale of measurement.

1. Determine the point estimate, and then construct the 95% confidence interval for μ7 =
the average attitude toward Springdale Mall.

2. Repeat part (a) for μ8 and μ9, the average attitudes toward Downtown and West Mall,
respectively.

2. Given the breakdown of responses for variable 26 (sex of respondent), determine the point
estimate and then construct the 95% confidence interval for p26 = the population proportion of
males.

3. Given the breakdown of responses for variable 28 (marital status of respondent), determine the
point estimate and then construct the 95% confidence interval for p28 = the population
proportion in the “single or other” category.

4. Assume the managers have requested estimates of the mean attitudes towards each mall with a
margin of error of 0.05 for each. If the managers want to have 95% confidence that the sample
mean will fall within this margin of error, how large should the sample size be for each mall?

1.The point estimate and the 95% confidence interval for μ7 = the average attitude toward Springdale
Mall is

The point estimate
x

x

n




standard deviation
 

2

1

x x
SD

n









The 95% confidence interval for mean average attitude is
0.05/ 2, 1n

SD
x t

n


 

Count 150

Average 3.813

Standard
deviation 1.064

Standard error 0.087

t-significant value
at 5% 1.976

Point estimate 3.813

95% Lower limit 3.642

95% upper limit 3.985

2. The point estimate and the 95% confidence interval for μ8 = the average attitude toward Down town
Mall is

The point estimate
x

x
n




standard deviation
 
2
1
x x
SD
n








The 95% confidence interval for mean average attitude is 0.05/ 2, 1n
SD

x t
n


 

Count 150

Average 3.407

Standard deviation 1.124

Standard error 0.092

t-significant value at 5% 1.976

Point estimate 3.407

95% Lower limit 3.225

95% upperlimit 3.588

3. The point estimate and the 95% confidence interval for μ9 = the average attitude toward West Mall is

The point estimate
x
x
n



standard deviation
 
2
1
x x
SD
n





The 95% confidence interval for mean average attitude is 0.05/ 2, 1n
SD
x t
n

 

Count 150

Average 3.287

Standard
deviation 1.183

Standard error 0.097

t-significant value
at 5% 1.976

Point estimate 3.287

95% Lower limit 3.096

95% upper limit 3.478

II The point estimate and the 95% confidence interval for p26 = the population proportion of males is

Point estimate
No.of Males

p
n



95% confidence interval for population proportion of males is
 

0.05/2

1p p
p Z

n




Count 150

No.of Males 64

No.of Females 86

Proportion of
Males 0.427

Z value at
0.05/2 level 1.960

Standard
error 0.040

Point
estimate 0.427

95%
confidence
interval lower
limit 0.348

95%
confidence
interval upper
limit 0.506

III
The point estimate and the 95% confidence interval for p28 = the population proportion in the “single or
other” category is

Point estimate
No.of single or other

p
n


95% confidence interval for population proportion single or other is
 

0.05/2
1p p
p Z
n



Count 150

NO. of Single
or other 83

No.of
Married 67

Proportion of
single or
other 0.553

Z value at
0.05/2 level 1.960

Standard
error 0.041

Point
estimate 0.553

95%
confidence
interval
lower limit 0.474

95%
confidence
interval
upper limit 0.633

IV.
The estimated sample sizes for each of the sample with the margin of error 0.05 with 95% confidence is

2

0.05/ 2
*

Sample size
Z SD

ME

 
  
 

SPRINGDALE DOWN TOWN WEST

Margin of error 0.05 0.05 0.05

Standard
deviation 1.064 1.124 1.183

Z-value at 0.05/2
level 1.96 1.96 1.96

Sample size 1741 1941 2152

The major shopping areas in the community of Springdale include Springdale Mall, West Mall, and the downtown area on Main Street. A telephone survey has been conducted to identify strengths and weaknesses of these areas and to find out how they fit into the shopping activities of local residents. The 150 respondents were also asked to provide information about themselves and their shopping habits. The data are provided in the file SHOPPING. The variables in the survey can be found in the file CODING.
In this section, hypothesis testing will be used to examine the significance of some of the survey results. From a managerial perspective, the information gained from this database exercise would be useful in determining whether the mean attitude toward a given area differed significantly from the “neutral” level, in observing and comparing the mean attitude scores exhibited toward the three shopping areas, and in examining the strengths and weaknesses of a manager’s shopping area relative to the other two areas. If such a study were repeated from time to time, management could observe the extent to which the overall attitude toward their shopping area was becoming stronger or weaker in the eyes of the consumer, thus helping them select business strategies that could retain and take advantage of their area’s strengths and either correct or minimize the effect of its weaknesses.
1.Item C in the description of the data collection instrument lists variables 7, 8, and 9, which represent the respondent’s general attitude toward each of the three shopping areas. Each of these variables has numerically equal distances between the possible responses, and for purposes of analysis they may be considered to be of the interval scale of measurement.

a. Calculate the mean attitude score toward the Springdale Mall shopping area and compute the sample standard deviation. Does this area seem to be well regarded by the respondents?

The mean attitude score toward the Springdale Mall is 3.6467.

This mean value puts the attitude toward Springdale Mall just over midway between “Neutral” and “Like”, so it would be a stretch to consider this mall to be “well regarded”. The rating is actually somewhat better than “indifferent”.
b. In a two-tail test at the 0.10 level, compare the sample mean calculated in part (a) to the 3.0 “neutral” value. Use the null hypothesis and the alternative hypothesis .

The hypotheses are:

Null:

Alternative:
The critical values for a two-tailed test at α = 0.10 are ±1.645.
The test statistic is 6.5436.
Since the test statistic is greater than the positive critical value, the decision is to reject the null hypothesis.

There is sufficient evidence at the 0.10 level of significance to support a claim that the population mean attitude toward Springdale Mall is different from 3.0.
c. Generate the 90% confidence interval for the population mean and use this interval in verifying the conclusion obtained in part (b).
The 90% confidence interval for the population mean is (3.4841, 3.8092).
Since the 90% confidence interval does not include the value 3.0, a claim that the population mean attitude toward Springdale Mall is different than 3.0 is
supported.
d. What is the p-value associated with the hypothesis test?
The p-value is 0.0000.
e. Repeat parts (a) through (d) for the Downtown shopping area.
The mean attitude score toward the Downtown shopping area is 3.3933.
This mean value puts the attitude toward Downtown shopping area just over the “Neutral” level. While the attitude is positive, it is not necessarily sufficiently different from the “neutral” level to be considered “well regarded”.
The hypotheses are:

Null:

Alternative:
The critical values for a two-tailed test at α = 0.10 are ±1.645.
The test statistic is 4.3612.
Since the test statistic is greater than the positive critical value, the decision is to reject the null hypothesis.

There is sufficient evidence at the 0.10 level of significance to support a claim that the population mean attitude toward the Downtown shopping area is different
from 3.0.

The 90% confidence interval for the population mean is (3.2450, 3.5417).
Since the 90% confidence interval does not include the value 3.0, a claim that the population mean attitude toward the Downtown shopping area is different than
3.0 is supported.
The p-value is 0.0000.

f. Repeat parts (a) through (d) for the West Mall shopping area.
The mean attitude score toward the West Mall shopping area is 3.2333.
This mean value puts the attitude toward West Mall shopping area just over the “Neutral” level. While the attitude is positive, it is not necessarily sufficiently different from the “neutral” level to be considered “well regarded”.
The hypotheses are:

Null:

Alternative:
The critical values for a two-tailed test at α = 0.10 are ±1.645.
The test statistic is 2.4373.
Since the test statistic is greater than the positive critical value, the decision is to reject the null hypothesis.

There is sufficient evidence at the 0.10 level of significance to support a claim that the population mean attitude toward the West Mall shopping area is different
from 3.0.
The 90% confidence interval for the population mean is (3.0759, 3.3908).
Since the 90% confidence interval does not include the value 3.0, a claim that the population mean attitude toward the West Mall shopping area is different than
3.0 is supported.
The p-value is 0.0148.

2.If Springdale Mall offered exactly the same benefits as the other two shopping areas, we would expect exactly one-third of those who expressed an opinion to select it as the area best fitting the description for variable number 10 (“Easy to return/exchange goods”). In testing whether Springdale Mall differs significantly from this expected proportion, the null and alternative hypotheses will be . Carry out the following analyses for Springdale Mall:
a. Analyzing the data for variable 10, use the preceding null and alternative hypotheses in testing the null hypothesis at the 0.05 level.
After removing all of the “4” responses (“4” = “No Opinion”), there are a total of 117 responses, and 60 that indicate a preference for Springdale Mall.

The sample proportion is:
The hypotheses are:

Null:

Alternative:
The critical values for a two-tailed test with α = 0.05 are ±1.96.
The test statistic is:

Since the test statistic is greater than the positive critical value, the decision is to
reject the null hypothesis.
There is sufficient evidence at the 0.05 level of significance to support a claim that the proportion who prefer the Springdale Mall differs from 0.33.
b. Determine the p-value for the test conducted in part (a).
The p-value for this test is 0.0000.

c. Repeat parts (a) and (b) for variables 11–17.
For the remaining variables, the hypotheses and critical values remain the same.
The tests and the corresponding results are:
Variable 11 (High quality of goods):
Test statistic value: 6.3056
Decision: Reject the null hypothesis
Conclusion: The proportion who prefer Springdale Mall in this category differs from 0.33.
Variable 12 (Low Prices)

Test statistic value: -3.1523
Decision: Reject the null hypothesis
Conclusion: The proportion who prefer Springdale Mall in this category differs from 0.33.
Variable 13 (Good variety of sizes/styles)
Test statistic value: 9.9551
Decision: Reject the null hypothesis
Conclusion: The proportion who prefer Springdale Mall in this category differs from 0.33.
Variable 14 (Sales staff helpful/friendly)
Test statistic value: 5.0786
Decision: Reject the null hypothesis
Conclusion: The proportion who prefer Springdale Mall in this category differs from 0.33.
Variable 15 (Convenient shopping hours)
Test statistic value: 7.6580
Decision: Reject the null hypothesis
Conclusion: The proportion who prefer Springdale Mall in this category differs from 0.33.
Variable 16 (Clean stores and surroundings)
Test statistic value: 10.2075
Decision: Reject the null hypothesis
Conclusion: The proportion who prefer Springdale Mall in this category differs from 0.33.
Variable 17 (A lot of bargain sales)
Test statistic value: 0.7792
Decision: Fail to reject the null hypothesis
Conclusion: The proportion who prefer Springdale Mall in this category is not different from 0.33.
d. Based on the preceding analyses, identify the principal strengths and weaknesses exhibited by Springdale Mall.
The analysis shows that Springdale Mall is doing well compared to the other malls in all categories except for Variable 12 (“Low Prices”) and Variable 17 (“A lot of bargain sales).
In the case of the “low prices” category, the hypothesis test shows that the proportion who selected Springdale Mall for this category is different from 0.33, but the negative test statistic value (-3.1523) indicates that Springdale is not doing well in this particular category, as their proportion of votes is well below 0.33.
In the case of the “bargain sales” category, the hypothesis test shows that the proportion who preferred Springdale Mall in this category is not significantly different from 0.33. This may mean that Springdale Mall is doing just as well as the other malls in this category, and may represent an opportunity for the mall to distance itself from its competitors. Of course, depending on the type of stores in the mall, being known for “bargain sales” may not fit with the mall’s overall image.

In all of the other categories, Springdale Mall’s proportion of votes exceeded 0.33, and the test statistic was positive, indicating that Springdale Mall outperformed its competitors in these categories.

p̂ =
x
n
=
60
117

= 0.5128

ˆ
p=
x
n
=
60
117
=0.5128

H0 : p = 0.33

H
0
:p=0.33

H1 : p ≠ 0.33

H
1
:p¹0.33

z =
x − np
p 1− p( )n

z =
60− 117( ) 0.33( )
0.33( ) 1− 0.33( ) 117( )

z = 4.2056

z=
x-np
p1-p
()
n
z=
60-117
()
0.33
()
0.33
()
1-0.33
()
117
()
z=4.2056

H0 :µ = 3.0

H
0
:µ=3.0

H1 :µ ≠ 3.0

H
1
:µ¹3.0