**T-Test Hypothesis **

**SPSS assignment**** **

[1] **Refer to the Softdrinks Corrected Database file and questionnaire for answering the following questions. **

- a) In today’s world of obesity, which demographic and psychographic variables are the best forecasters of the percentage of sugared soft drinks a person drinks?
- b) Do Caucasian Americans drink significantly more soda on average than African Americans? c) Investigate how many people in the sample are brand loyal? Assume that people are brand loyal if they drink a simple majority (more than 50%) of their favorite drinks.

[2**] Please refer to the Discover database and questionnaire for answering the following questions.**

- a) Please test that the mean overall satisfaction and the mean “value of the Discover card” to customers are significantly above 3.5.
- b) Is the likelihood of a person keeping the card related significantly more to the value they have for the card or to their satisfaction with the card? Is the likelihood of a person recommending the card related more to its value or to the satisfaction perceived by the customer?
- c) Is there a significant difference between how consumer’s rate courtesy and friendliness of a customer service agent? Is there a significant difference between how consumers perceive the professionalism and efficiency of handling the customer service call? Finally, is there a significant difference between the customer’s ratings of the rep’s concern for her needs and the reps’ ability to make the customer feel important? Does it make sense to measure these three questions using two survey items each?

**Solution**

[1]

(a)

The demographic and psychographic factors that may be the forecasters of the percentage of sugared soft drinks a person drinks are – ethnic orientation, gender, classification,2 and age.

Between-Subjects Factors |
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Value Label | N | ||

ethnic orientation | 1 | caucasian | 362 |

2 | African-American | 59 | |

gender | 0 | male | 188 |

1 | female | 233 | |

classification | 1 | freshman | 38 |

2 | sophomore | 49 | |

3 | junior | 75 | |

4 | senior | 214 | |

5 | grad student | 45 | |

age | 1 | 0-18 years | 19 |

2 | 19-20 years | 92 | |

3 | 21-22 years | 162 | |

4 | 23-25 years | 77 | |

5 | 26-30 | 17 | |

6 | over 30 | 54 |

We apply a multiple regression to investigate the statistically significant predictors (among these 4 demographic and psychographic factors) of the percentage of sugared soft drinks a person drinks.

Variables Entered/Removed^{a} |
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Model | Variables Entered | Variables Removed | Method |

1 | age, ethnic orientation, gender, classification^{b} |
. | Enter |

a. Dependent Variable: % soft drinks with sugar | |||

b. All requested variables entered. |

Model Summary |
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Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |

1 | .246^{a} |
.061 | .052 | 36.256 |

a. Predictors: (Constant), age, ethnic orientation, gender, classification |

ANOVA^{a} |
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Model | Sum of Squares | df | Mean Square | F | Sig. | |

1 | Regression | 35325.570 | 4 | 8831.392 | 6.719 | .000^{b} |

Residual | 546826.744 | 416 | 1314.487 | |||

Total | 582152.314 | 420 | ||||

a. Dependent Variable: % soft drinks with sugar | ||||||

b. Predictors: (Constant), age, ethnic orientation, gender, classification |

Coefficients^{a} |
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Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | 95.0% Confidence Interval for B | |||

B | Std. Error | Beta | Lower Bound | Upper Bound | ||||

1 | (Constant) | 88.242 | 8.730 | 10.108 | .000 | 71.082 | 105.401 | |

ethnic orientation | 9.406 | 5.101 | .088 | 1.844 | .066 | -.620 | 19.432 | |

gender | -14.822 | 3.612 | -.198 | -4.104 | .000 | -21.921 | -7.722 | |

classification | -1.194 | 1.759 | -.036 | -.678 | .498 | -4.652 | 2.265 | |

age | -4.082 | 1.472 | -.147 | -2.773 | .006 | -6.976 | -1.189 | |

a. Dependent Variable: % soft drinks with sugar |

The factors ethnic orientation and classification are not statistically significant in predicting the percentage of sugared soft drinks a person drinks as their p-value is higher than 0.05

The p-value for the factors gender and age are very low and hence they are statistically significant in predicting the percentage of sugared soft drinks a person drinks.

Hence, the demographic and psychographic factors which are the best predictors of the percentage of sugared soft drinks a person drinks are – gender and age.

(b)

Our aim is to investigate whether Caucasian Americans drink significantly more soda on average than African Americans. We formulate the following hypothesis:

H0: On average, the Caucasian Americans and African Americans drink the same amount of soda.

H1: The Caucasian Americans drink significantly more soda on average than African Americans.

To test this hypothesis, we perform an independent samples t-test.

Group Statistics |
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ethnic orientation | N | Mean | Std. Deviation | Std. Error Mean | |

Average weekly consumption of soft drinks (12 oz cans) | caucasian | 364 | 9.90 | 9.049 | .474 |

African-American | 59 | 7.86 | 5.438 | .708 |

Independent Samples Test |
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Levene’s Test for Equality of Variances | t-test for Equality of Means | ||||||||

F | Sig. | t | df | Sig. (2-tailed) | Mean Difference | 95% Confidence Interval of the Difference | |||

Lower | Upper | ||||||||

Average weekly consumption of soft drinks (12 oz cans) | Equal variances assumed | 2.167 | .142 | 1.679 | 421 | .094 | 2.037 | -.347 | 4.421 |

Equal variances not assumed | 2.390 | 117.957 | .018 | 2.037 | .349 | 3.724 |

The p-value for the Levene’s test for equality of variances of the samples is 0.142

Hence, we fail to reject the assumption of the equality of variances of the two samples.

Assuming the equal variances of the average weekly consumption of soft drinks in Caucasian and African Americans, the p-value for the two-test t-test is 0.094

So, the p-value for the one-sided independent samples t-test that we have performed to test the null hypothesis is 0.094 / 2 = 0.047, which is statistically significant.

Hence, we reject the null hypothesis H0 against the alternate hypothesis H1.

Thus, we conclude that Caucasian Americans drink significantly more soda on average than African Americans.

(c)

% of soft drinks consumed are favorite soft drink Stem-and-Leaf Plot

Frequency Stem & Leaf

4.00 0 . 0005

3.00 1 . 005

11.00 2 . 00000055555

9.00 3 . 000003344

16.00 4 . 0000000000000555

83.00 5 . 00000000000000000000000000000000000000000000000000000000000000000000000000000000005

34.00 6 . 0000000000000000000000000000000555

79.00 7 . 0000000000000000000000000005555555555555555555555555555555555555555555555555555

83.00 8 . 00000000000000000000000000000000000000000000000000000000000000000005555555555555599

92.00 9 . 00000000000000000000000000000000000000000000000000244455555555555555555555555555577788888899

52.00 10 . 0000000000000000000000000000000000000000000000000000

Stem width: 10

Each leaf: 1 case(s)

From the stem and leaf plot for the percentage of favorite soft drinks consumed, we observe that 341 people in the sample drink a simple majority (more than 50%) of their favorite drinks and 82 people in the sample drink 50% of their favorite drinks.

Hence, 341 people in the sample are brand loyal.

[2]

(a)

We want to test that the mean overall satisfaction and the mean “value of the Discover card” to customers are significantly above 3.5.

To test that the mean overall satisfaction is significantly above 3.5, we test the following hypothesis:

H0: The mean overall satisfaction is equal to 3.5

H1: The mean overall satisfaction is above 3.5

To test that the mean value of the Discover card is significantly above 3.5, we test the following hypothesis:

H0: The mean value of the Discover card is equal to 3.5

H1: The mean value of the Discover card is above 3.5

To test these two hypotheses, we perform a one-sample t-test.

One-Sample Statistics |
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N | Mean | Std. Deviation | Std. Error Mean | |

overall satisfaction | 244 | 3.9877 | 1.00403 | .06428 |

value of the card | 244 | 3.3852 | 1.20661 | .07725 |

One-Sample Test |
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Test Value = 3.5 | ||||||

t | df | Sig. (2-tailed) | Mean Difference | 95% Confidence Interval of the Difference | ||

Lower | Upper | |||||

overall satisfaction | 7.588 | 243 | .000 | .48770 | .3611 | .6143 |

value of the card | -1.486 | 243 | .139 | -.11475 | -.2669 | .0374 |

The p-value of the one-sided t-test to test the hypothesis defined for the mean overall satisfaction is very low. So, we reject the null hypothesis that the mean overall satisfaction is equal to 3.5 against the alternative hypothesis that the mean overall satisfaction is above 3.5

Hence, we conclude that the mean overall satisfaction is significantly above 3.5

The p-value of the one-sided t-test to test the hypothesis defined for the mean value of the Discover card is 0.139 / 2 = 0.0695 which is more than 0.05. So, we fail to reject the null hypothesis that the mean value of the Discover card is equal to 3.5 against the alternative hypothesis that the mean value of the Discover card is above 3.5

Hence, we conclude that the mean value of the Discover card is not significantly above 3.5

(b)

We want to investigate if the likelihood of a person keeping the card related significantly more to the value they have for the card or to their satisfaction with the card and if the likelihood of a person recommending the card related more to its value or to the overall satisfaction perceived by the customer.

For this, we conduct a paired sample t-test for the 4 pairs of responses.

Paired Samples Statistics |
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Mean | N | Std. Deviation | Std. Error Mean | ||

Pair 1 | keeping card | 4.2213 | 244 | .81157 | .05196 |

value of the card | 3.3852 | 244 | 1.20661 | .07725 | |

Pair 2 | keeping card | 4.2213 | 244 | .81157 | .05196 |

overall satisfaction | 3.9877 | 244 | 1.00403 | .06428 | |

Pair 3 | recommending card | 3.2582 | 244 | 1.12717 | .07216 |

value of the card | 3.3852 | 244 | 1.20661 | .07725 | |

Pair 4 | recommending card | 3.2582 | 244 | 1.12717 | .07216 |

overall satisfaction | 3.9877 | 244 | 1.00403 | .06428 |

Paired Samples Correlations |
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N | Correlation | Sig. | ||

Pair 1 | keeping the card & value of the card | 244 | .564 | .000 |

Pair 2 | keeping card & overall satisfaction | 244 | .584 | .000 |

Pair 3 | recommending the card & value of the card | 244 | .632 | .000 |

Pair 4 | recommending card & overall satisfaction | 244 | .588 | .000 |

Paired Samples Test |
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Paired Differences | t | df | Sig. (2-tailed) | ||||||

Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | ||||||

Lower | Upper | ||||||||

Pair 1 | keeping card – value of the card | .83607 | 1.00501 | .06434 | .70933 | .96280 | 12.995 | 243 | .000 |

Pair 2 | keeping card – overall satisfaction | .23361 | .84543 | .05412 | .12700 | .34022 | 4.316 | 243 | .000 |

Pair 3 | recommending card – value of the card | -.12705 | 1.00423 | .06429 | -.25368 | -.00041 | -1.976 | 243 | .049 |

Pair 4 | recommending card – overall satisfaction | -.72951 | .97320 | .06230 | -.85223 | -.60679 | -11.709 | 243 | .000 |

The p-value for testing the equality of the responses – keeping the card and value of the card is extremely low, in the order of 10^{-29} whereas the p-value for testing the equality of responses – keeping the card and overall satisfaction is 0.000023

Hence, the likelihood of a person keeping the card is related significantly more to their overall satisfaction with the card than to the value they have for the card.

The p-value for testing the equality of the responses – recommending the card and value of the card is 0.049 whereas the p-value for testing the equality of responses – recommending the card and overall satisfaction is less than 0.001

Hence, the likelihood of a person keeping the card is related significantly more to the value they have for the card than to their overall satisfaction with the card.

(c)

We want to test if there is a significant difference between how consumer’s rate courtesy and friendliness of a customer service agent.

We want to test if there is a significant difference between how consumers perceive professionalism and efficiency of handling the customer service call.

Finally, we want to test if there is a significant difference between the customer’s ratings of the rep’s concern for her needs and the reps’ ability to make the customer feel important.

So, there are 3 pairs of responses:

- Courtesy and friendliness
- Professionalism and efficiency
- Concern and making feel important

For each of these three pairs, we need to test the following hypothesis:

H0: There is no difference between the ratings of the two responses of the pair.

H1: There is a difference between the ratings of the two responses of the pair.

To test these, we have to apply a two-sided paired samples t-test for the 3 pairs of responses.

Paired Samples Statistics |
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Mean | N | Std. Deviation | Std. Error Mean | ||

Pair 1 | courtesy | 4.3730 | 244 | .82418 | .05276 |

friendliness | 4.2254 | 244 | .90858 | .05817 | |

Pair 2 | professionalism | 4.3852 | 244 | .85053 | .05445 |

efficiency | 4.2828 | 244 | .95039 | .06084 | |

Pair 3 | concern | 3.8484 | 244 | 1.19221 | .07632 |

important feeling | 3.3238 | 244 | 1.28848 | .08249 |

Paired Samples Correlations |
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N | Correlation | Sig. | ||

Pair 1 | courtesy & friendliness | 244 | .750 | .000 |

Pair 2 | professionalism & efficiency | 244 | .628 | .000 |

Pair 3 | concern & important feeling | 244 | .646 | .000 |

Paired Samples Test |
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Paired Differences | t | df | Sig. (2-tailed) | ||||||

Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | ||||||

Lower | Upper | ||||||||

Pair 1 | courtesy – friendliness | .14754 | .61760 | .03954 | .06966 | .22542 | 3.732 | 243 | .000 |

Pair 2 | professionalism – efficiency | .10246 | .78157 | .05004 | .00390 | .20102 | 2.048 | 243 | .042 |

Pair 3 | concern – important feeling | .52459 | 1.04793 | .06709 | .39244 | .65674 | 7.820 | 243 | .000 |

For all of these two-sided paired two-sample t-test, the p-values are less than 0.05 and are thus statistically significant.

Hence, the null hypothesis of no difference between the ratings of the two responses of the pair is rejected against the alternative hypothesis of significant difference between the ratings of the two responses of the pair for all the three pairs of responses.

So, we conclude that there is a statistically significant difference between how consumer’s rate courtesy and friendliness of a customer service agent.

We conclude that there is a statistically significant difference between how consumers perceive professionalism and efficiency of handling the customer service call.

We conclude that there is a statistically significant difference between the customer’s ratings of the rep’s concern for her needs and the reps’ ability to make the customer feel important.

It does not make sense to measure these three questions using two survey items each.