Question

Difference in population means confidence interval

Answer 1

There are many situations where it is of interest to compare two groups with respect to their mean scores on a continuous outcome. For example, we might be interested in comparing mean systolic blood pressure in men and women, or perhaps compare body mass index (BMI) in smokers and non-smokers. Both of these situations involve comparisons between two independent groupsThe confidence interval will be computed using either the Z or t distribution for the selected confidence level and the standard error of the point estimate. The use of Z or t again depends on whether the sample sizes are large (n1> 30 and n2 > 30) or small. The standard error of the point estimate will incorporate the variability in the outcome of interest in each of the comparison groups. If we assume equal variances between groups, we can pool the information on variability (sample variances) to generate an estimate of the population variability. Therefore, the standard error (SE) of the difference in sample means is the pooled estimate of the common standard deviation (Sp) (assuming that the variances in the populations are similar) computed as the weighted average of the standard deviations in theFor both large and small samples Sp is the pooled estimate of the common standard deviation (assuming that the variances in the populations are similar) computed as the weighted average of the standard deviations in the samples.

The table below summarizes data n=3539 participants attending the 7th examination of the Offspring cohort in the Framingham Heart Study. 

 

Men

Women

Characteristic

N



s

n



s

Systolic Blood Pressure

1,623

128.2

17.5

1,911

126.5

20.1

Diastolic Blood Pressure

1,622

75.6

9.8

1,910

72.6

9.7

Total Serum Cholesterol

1,544

192.4

35.2

1,766

207.1

36.7

Weight

1,612

194.0

33.8

1,894

157.7

34.6

Height

1,545

68.9

2.7

1,781

63.4

2.5

Body Mass Index

1,545

28.8

4.6

1,781

27.6

5.9

Suppose we want to calculate the difference in mean systolic blood pressures between men and women, and we also want the 95% confidence interval for the difference in means. The sample is large (> 30 for both men and women), so we can use the confidence interval formula with Z. Next, we will check the assumption of equality of population variances. The ratio of the sample variances is 17.52/20.12 = 0.76, which falls between 0.5 and 2, suggesting that the assumption of equality of population variances is reasonable.

First, we need to compute Sp, the pooled estimate of the common standard deviation.



Substituting we get



which simplifies to



Notice that for this example Sp, the pooled estimate of the common standard deviation, is 19, and this falls in between the standard deviations in the comparison groups (i.e., 17.5 and 20.1). Next we substitute the Z score for 95% confidence, Sp=19, the sample means, and the sample sizes into the equation for the confidence interval.

Answer 2

Answer:

A confidence interval (C.I.) specifies a set of possible values for a given parameter.

Explanation:

Step 2

Effect of having different levels of confidence in estimation of population mean affects margin or error(M.O.E). As increasing the size of C.I increases the possibility that the parameter value lies in the interval which also broadens the M.O.E  whereas decreasing the size of C.I decreases the possibility that the parameter value lies in the interval which also decreases the M.O.E.