PHE5IST Biostatistics

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02-07-2021

Question:

Below are estimates of the daily intake of calcium (in milligrams) for 38 women between the ages of 51 and 80 years who participated in a study of women’s bone health:
808 882 1062 970 909 802 374 416 784 997
651 716 438 1420 1525 948 1050 976 572 403
626 774 1253 549 1325 446 465 1769 671 696
1156 684 1933 748 1203 2433 1255 1100

a. Using the above histogram describe the distribution of calcium intakes for these women.
b. Calculate the mean, median, range, standard deviation, variance and the standard error.
c. Find a 95% confidence interval for the mean. Interpret this confidence interval.
d. Suppose that the recommended daily allowance (RDA) of calcium for women in this age is 1200 milligrams (this value is changed from time to time on the basis of the statistical analysis of new data). Express the mean and median of the results from b in terms of percent of the RDA.
e. Divide each intake by the RDA, multiple by 100, and compute the 95% confidence interval from the transformed data.
f. Verify that you can obtain the same results by similarly transforming the interval you calculated in c.

A rare congenital disease, Everley's syndrome, generally causes a reduction in concentration of blood sodium. This is thought to provide a useful diagnostic sign as well as a clue to the efficacy of treatment. Over the course of seven years, 20 cases of Everley’s syndrome were diagnosed at a London teaching hospital. The patients were all aged
between 20 and 44 and agreed to take part in an experimental treatment program.At diagnosis, the blood sodium levels of these 20 cases were:
110 95 120 112 106 109 98 152 111 105
96 97 114 116 87 142 95 111 120 122

a. Assuming that blood sodium concentration is normally distributed what is the 95% confidence interval within which the mean of the total population of such cases may be expected to lie.
After 12 months of the experimental treatment program, the blood sodium levels of these 20 cases were (in order):
108 126 119 106 116 122 114 148 120 110
127 142 130 126 114 134 121 135 120 126
b. Is the experimental method of treatment effective at raising the blood sodium levels of patients with Everley’s syndrome Justify your answer using statistical methods and language.

Answer:

Description of the distribution of calcium intake.

The distribution is normal with a tail towards the right. The distribution is also less kurtic since it does not have a sharp top. It is also characterized by a gap in between the bars.

Descriptive statistics

Column1
Mean	941.8157895
Standard Error	72.77628824
Median	845
Mode	#N/A
Standard Deviation	448.6231703
Sample Variance	201262.7489
Kurtosis	2.323888221
Skewness	1.333427894
Range	2059
Minimum	374
Maximum	2433
Sum	35789
Count	38

Table 1 (generated from excel)

95% C.I for the mean consumption of calcium.

C.I is calculated as below,

The formula is = Mean = 941.81578

n = 38

Z value for 95% from normal tables = 1.65

941.82

Mean and median as a percentage of RDA.

If RDA = 1200

Mean = 941.82

Median = 845

Mean as a percentage of RDA =

Transformed data descriptive statistics.

Column1
Mean	78.48464912
Standard Error	6.064690687
Median	70.41666667
Mode	#N/A
Standard Deviation	37.38526419
Sample Variance	1397.657979
Kurtosis	2.323888221
Skewness	1.333427894
Range	171.5833333
Minimum	31.16666667
Maximum	202.75
Sum	2982.416667
Count	38

Table 2

C.I is calculated as below,

The formula is = Mean = 78.484

n = 38

Z value for 95% from normal tables = 1.65 78.484

Since we divided the intake by 1200 and multiplied by 100, we can verify that we can obtain the same results by performing the inverse. That is, we multiply by 1200 then divide by 100.

Taking the mean,

Same to the interval value,

95% confidence for the mean of blood sodium concentration

Column1
Mean	110.9
Standard Error	3.518298034
Median	110.5
Mode	95
Standard Deviation	15.73430714
Sample Variance	247.5684211
Minimum	87
Maximum	152
Sum	2218
Count	20

Table 3

The formula is Mean = 110.9

n = 20

Z value for 95% from normal tables = 1.65 110.9

Is the treatment effective after 12 months?

To establish the effect, we perform a t-test of before and after to see whether the change was significant.

Our null and alternative hypothesis is as below;

H₀: There is no significant difference in the blood sodium levels before and after the treatment.

Versus

H₁: There is a significant difference in the blood sodium levels before and after the treatment.

Level of significance is 0.05

Paired Samples Test
	Paired Differences					t	df	Sig. (2-tailed)
	Mean	Std. Deviation	Std. Error Mean	95% Confidence Interval of the Difference
				Lower	Upper
before - after	-12.30000	14.55697	3.25504	-19.11287	-5.48713	-3.779	19	.001

Table 4

So as to arrive at the conclusion, we should compare the p-value and significance level.

In this case the p-value is less than the level of significance (.001<.05), therefore we reject the null hypothesis and accept the alternative. So we conclude that there is a significant difference in the blood sodium levels before and after the treatment.