# Confidence Interval

Confidence Interval

In statistics, confidence interval is used to describe the level of uncertainty that a sample estimate of a population parameter is associated with. Also, it describes the uncertainty that a sampling method is associated with in a situation where a sampling method is used to select different samples and to estimate different intervals. Some estimates do include true parameters of population while some do not.

• Construct a 99% confidence interval estimate of the mean body mass index for men.
 BMI MALE FEMALE 23.8 19.6 23.2 23.8 24.6 19.6 26.2 29.1 23.5 25.2 24.5 21.4 21.5 22.0 31.4 27.5 26.4 33.5 22.7 20.6 27.8 29.9 28.1 17.7 25.2 24.0 23.3 28.9 31.9 37.7 33.1 18.3 33.2 19.8 26.7 29.8 26.6 29.7 19.9 31.7 27.1 23.8 23.4 44.9 27.0 19.2 21.6 28.7 30.9 28.5 28.3 19.3 25.5 31.0 24.6 25.1 23.8 22.8 27.4 30.9 28.7 26.5 26.2 21.2 26.4 40.6 32.1 21.9 19.6 26.0 20.7 23.5 26.3 22.8 26.9 20.7 25.6 20.5 24.2 21.9 MEAN 25.9975 25.74 STD DEV 3.430742 6.16557

Confidence interval is given by

The mean of BMI values for males = 25.9975.

The standard deviation of BMI values for males = 3.4307.

The sample size is 40.

For a 99% confidence interval, the critical value of z is given by the excel function NORMSINV(0.01/2) = 2.5758.

Standard error of mean
= standard deviation/ sqrt(sample size)
= 3.4307/sqrt(40)
= 0.54.

Margin of error
= critical value of z*Standard error of mean
= 2.5758 * 0.54
= 1.3909.

So, the lower limit of a99% confidence interval estimate of the mean body mass index for males is:

= 25.9975 – 2.5758*3.4307/sqrt(40)

=24.6066

The upper limit of a99% confidence interval estimate of the mean body mass index for males is:

= 25.9975 + 2.5758*3.4307/sqrt(40)

=27.3884

Hence, a 99% confidence interval estimate of the mean body mass index for males is (24.6066, 27.3884).

• Construct a 99% confidence interval estimate of the mean body mass index for women.

Confidence interval is given by

The mean of BMI values for females = 25.74.

The standard deviation of BMI values for females = 6.1656.

The sample size is 40.

For a 99% confidence interval, the critical value of z is given by the excel function NORMSINV(0.01/2) = 2.5758.

So, the lower limit of a 99% confidence interval estimate of the mean body mass index for females is:

= 25.74 – 2.5758*6.1656/sqrt(40)

=23.2289.

The upper limit of a99% confidence interval estimate of the mean body mass index for females is:

= 25.74 + 2.5758*6.1656/sqrt(40)

=28.2511

Hence, a 99% confidence interval estimate of the mean body mass index for females is (23.2289, 28.2511).

• Compare and interpret the results. It is known that men have a mean weight that is greater than the mean weight for women, and the mean height of men is greater than the mean height of women, but do men also have a mean body mass index that is greater > than the mean body mass index of women?

To test if the BMI of male is greater than female we have to use T-test for difference in means.

Null hypothesis H0: BMI of male = BMI of female.

Alternate hypothesis, H1: BMI of male > BMI of female.

For testing difference in mean we will have to conduct 1 tail t test for difference in mean.

 BMI for Males BMI for Females Mean 25.9975 25.74 Variance 11.76999359 38.01425641 Observations 40 40 Pearson Correlation -0.064993278 Hypothesized Mean Difference 0 df 39 t Stat 0.224692134 P(T<=t) one-tail 0.411695951 t Critical one-tail 1.684875122

Since, p-value = 0.411, which is greater than alpha (0.05), I fail to reject null hypothesis at 5% level of significance and conclude that there is no significant difference between BMI of males and females.

References

Frederic, P. M. A. F. V. J. M. (2009). Confidence interval. Place of publication not identified: Vdm Pub. House.

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