1)You get a job as a traveling salesperson for Callahan Brake pads. You try to sell your firstclient on the idea that Callahan Brake pads are superior in quality. The client agrees but claims that your product is too expensive. He provides the following figures to support his claim. Callahan Brakes pads cost $15 per pair. The average cost of your 5 leading competitors $13.62 with s = 1.09. Conduct a one-sample hypothesis test (alpha = .05) to determine if the cost of Callahan Brake pads are in fact different from average.
Ho: = 15tcrit = 2.776
Ha: 15
tobs = (M - ) / [s / sqrt(n)]
= 13.62 - 15 / (1.09 / sqrt(5)
= -1.38 / (1.09 / 2.24)
= -1.38 / .49
= -2.82
Because tobs falls in the rejection region, we would reject the null: t(4) = -2.82, SEM = .49, p < .05. Unfortunately, this means that Callahan brake pads are more expensive than average.
2) Use SPSS to determine if Amherst students are eating enough fruit. Assume that the USRDA is 3 pieces of fruit. Set alpha = .01.
One-Sample StatisticsN / Mean / Std. Deviation / Std. Error Mean
Fruits / 124 / 2.102 / 2.1834 / .1961
One-Sample Test
Test Value = 3
t / df / Sig. (2-tailed) / Mean Difference / 95% Confidence Interval of the Difference
Lower / Upper
Fruits / -4.578 / 123 / .000 / -.8976 / -1.286 / -.509
The data indicate that the average Amherst student eats 2.102 pieces of fruit per day. The p-value (.000) was less than alpha (.05). The hypothesis test indicates that this is significantly below the US RDA of 3 servings of fruit per day: t (123) = -4.578, SEM 0.20, p < .05. Apparently, y’all need to eat more fruit!
3) You and Biff are playing a heated game of Jacks, when the conversation invariably turns to who is the superior player. You both know enough statistics to know that one game won't settle the matter completely. So, you each play six games, you pick up 5, 4, 8, 4, 5, and 6 jacks. Biff picks up 4, 4, 5, 3, 4, and 5 jacks. Conduct a two-sample t-test (alpha = .05) to determine who is the better jackster.Do the calculations by hand, but you may check your work with SPSS.
You / Biff5, 4, 8, 4, 5, 6 / 4, 4, 5, 3, 4, 5
(x) = 32 / (x) = 25
(x2) = 182 / (x2) = 107
Mean = 5.33 / Mean = 4.17
s = 1.51 / s = .75
Ho: y = btcrit = 2.228
Ho: y b
s2p= n1-1 (s21) + n2-1 (s22) / n1 + n2 - 2
= (6-1)(1.512) + (6-1)(.752) / 6 + 6- 2
= [(5) (2.267) + (5) (.567)] / 10
= 11.33 + 2.83 / 10
= 14.167 / 10= 1.42
or
s2p= SS1 + SS2 / df1 + df2
= 11.33 + 2.83 / 5 + 5
= 14.167 / 10= 1.42
SE = s2p / n1 + s2p / n1
= 1.42 / 6+ 1.42 / 6
= .472= .687
tobs = (Meany - Meanb) / (SE)
= 5.33 - 4.17 / .687
= 1.16 / .687
= 1.69
Because tobs does not fall in the rejection region, we would fail to reject the null: t(10) = 1.69, SEM = .687, p> .05. This means that you do not have enough evidence to conclude that you are a better jackster than Biff.
4) Use SPSS to perform a two-sample t-test to determine if Amherst students consume different amounts of fruits and vegetables (hint: is this a paired test or an independent test). Set alpha = .01.
Paired Samples StatisticsMean / N / Std. Deviation / Std. Error Mean
Pair 1 / Fruits / 2.102 / 124 / 2.1834 / .1961
Veggies / 2.212 / 124 / 1.2920 / .1160
Paired Samples Test
Paired Differences
Mean / Std. Deviation / Std. Error Mean / 95% Confidence Interval / t / df / Sig. (2-tailed)
Lower / Upper
Pair 1 / Fruits - Veggies / -.1097 / 2.3765 / .2134 / -.5321 / .3128 / -.514 / 123 / .608
This is a paired t-test because each subject in the sample contributed two pieces of data (one for fruit, one for vegetables). The mean servings of fruit and vegetables are 2.102 and 2.212, respectively; thus, the mean difference score was -0.1097. The paired t-test revealed no significant difference in food consumption as the p-value (.608)was greater than alpha (.05): t(123) = -0.514, SEM = 0.213, p .01. Thus, there is not enough evidence to conclude that Amherst students’ fruit and vegetable consumption differ from one another.
5) Use SPSS to determine whether females and males consume the same amount of vegetables? Set alpha to be an appropriate level.
Gender / N / Mean / Std. Deviation / Std. Error Mean
Veggies / Female / 87 / 2.218 / 1.2615 / .1352
Male / 37 / 2.197 / 1.3789 / .2267
Independent Samples Test
Levene's Test for Equality of Variances / t-test for Equality of Means
F / Sig. / t / df / Sig. (2-tailed) / Mean Difference / Std. Error Difference / 95% Confidence Interval of the Difference
Lower / Upper
Veggies / Equal variances / .693 / .407 / .083 / 122 / .934 / .0211 / .2546 / -.4829 / .5251
This should be an independent t-test because males and females represent two distinct groups. The mean number of servings of vegetables per day for the males and females were 2.197 and 2.218, respectively. The t-test revealed that the difference between the males and females is not significant: t(122) = 0.083, SEM = 0.2546, p .05. The results of our test suggest that there is not enough evidence to conclude that there are gender differences in vegetable consumption. Note: every other year that I have used this example, females have eaten more veggies than males and ‘real’ data confirms this gender difference).