Question

A recent study examined the effect of diet cola consumption on calcium levels in women. A sample of 16 healthy women aged 18 – 40 were randomly assigned to drink 24 ounces of either diet cola or water. Their urine was collected for three hours after ingestion of the beverage and calcium excretion (in mg) was measured. The researchers were investigating whether diet cola leaches calcium out of the system, which would increase the amount of calcium in the urine for diet cola drinkers. The data are given below and are stored in the ColaCalcium data set.

Diet Cola 50 62 48 55 58 61 58 56

Water 48 46 54 45 53 46 53 48

1. By hand, find 10 bootstrap samples of each and sketch a dotplot showing the means of the samples. Also indicate the mean of the original data and the samples.

2. Find 1000 bootstrap samples of each and take a screen shot of the resulting bootstrap distributions and paste them in the paper.

3. Does it appear that the mean calcium lost for women who drink diet cola is more than for women who drink water? Justify your answer.

4. Identify the Standard Error for each data set then construct a 95% confidence interval to estimate the size of the effect

Answer 1

1)

Bootstrap sampling with hands: We now re-sample with replacement from our sample to form what is known as bootstrap samples. Each bootstrap sample will have a size of 8, just like our original sample. Since we randomly selecting and then are replacing each value, the bootstrap samples may be different from the original sample and from each other.

	Original	Bootstrap resampling
N	Diet Cola	B1	B2	B3	B4	B5	B6	B7	B8	B9	B10
1	50	50	55	48	55	48	61	58	56	55	61
2	62	55	61	55	50	58	58	50	55	50	62
3	48	58	58	62	58	61	55	62	56	48	55
4	55	61	48	58	61	55	48	61	48	48	62
5	58	55	62	58	48	56	61	48	50	55	56
6	61	56	50	56	61	56	50	55	61	58	58
7	58	48	61	61	56	48	50	50	62	56	61
8	56	61	50	61	55	61	56	56	48	55	52
Mean	56	55.5	55.625	57.375	55.5	55.375	54.875	55	54.5	53.125	58.375

1 sample is the original data sample mean.

	Original	Bootstrap resampling
N	Water	B1	B2	B3	B4	B5	B6	B7	B8	B9	B10
1	48	54	54	46	45	48	46	53	53	48	53
2	46	45	46	54	46	46	48	54	54	45	54
3	54	53	53	45	53	53	53	53	54	46	48
4	45	53	53	53	46	45	53	46	46	46	48
5	53	46	46	46	48	54	46	48	45	53	46
6	46	53	53	53	54	53	54	46	48	54	53
7	53	48	48	48	54	46	45	53	56	53	46
8	48	46	46	48	53	46	48	45	55	45	45
Mean	49.125	49.75	49.875	49.125	49.875	48.875	49.125	49.75	51.375	48.75	49.125

2)

1000 bootstrap samples of diet cola from statkey

1000 bootstrap samples of water

3)

Carried test for difference in mean in statkey assuming both means are same. The right p vlaue = 0.025 which shows the result is significant thus we reject null hypothesis and accept alternate hypothesis. Thus we conclude that mean calcium is more in diet cola as compared to water.

4)

As calculated in plots above

The standard error of 1000 samples of diet cola = 0.678

Mean of 1000 samples of diet cola = 55.973

The standard error of 1000 samples of Water = 0.515

Mean of 1000 samples of water = 49.260

The sample mean diet cola = 56

the sample mean water = 49.125

s.e of sample diet cola= 1.742

s.e of sample water = 1.28

the standard deviation of diet cola = 4.928

the standard deviation of water = 3.642

The effect size of diet cola d =

= 56 - 55.973/4.928 = 0.005479

The effect size of water = 49.125 - 49.260/3.642 = -0.03706

Confidence interval(95%)

Diet Cola

56 +/- 2.63*4.928/sqrt(8) = 56 +/- 4.12 = [51.88, 60.12]

Water

49.125 +/-2.63*3.642/sqrt(8) = 49.125 +/- 3.04479 = [46.08021, 52.16979]