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.
