Posted: April 1st, 2015

Calculating Confidence Intervals

Calculating Confidence Intervals

Calculating Confidence Intervals

Problem 4.1:

The 95% confidence intervals for the mean of the variable weight

The significance of the t test is .000. This indicates that the sample mean is significantly different from the test value. The lower 95% confidence limit is 148.9287 kilograms and the upper 95% confidence limit is 168.5808 kilograms.

The 90% confidence intervals for the mean of the variable weight

The significance of the t test is .000. This indicates that the sample mean is significantly different from the test value. The lower 90% confidence limit is 150.5542 kilograms and the upper 95% confidence limit is 166.9553 kilograms.

The 99% confidence intervals for the mean of the variable weight

The significance of the t test is .000. This indicates that the sample mean is significantly different from the test value. The lower 99% confidence limit is 145.6621 kilograms and the upper 95% confidence limit is 171.8473 kilograms.

From the above analysis, it is evident that the 95% confidence intervals for the mean of the variable weight have a difference of 19.6521. The 90% confidence intervals for the mean of the variable weight have a difference of 16.4011. The 99% confidence intervals for the mean of the variable weight have a difference of 26.1852. This indicates that as the when the between the extremes of the confidence interval is increased, the likelihood that the mean is in that range so as to have increased the confidence in the estimation also increases (Downing & Clark, n.d.; Jackson, 2012).

Problem 4.2:

1. The proportion of each group of respondents

Proportion of respondents who smoke everyday

54,815/426,000 = 0.1287

Proportion of respondents who smoke some days

21,581/426,000 = 0.0507

Proportion of respondents who are former smokers

110,060/426,000 = 0.2584

Proportion of respondents who have never smoked

143,619/426,000 =0.3371

1. The confidence interval for the proportion of respondents who stated they are former smokers

Calculating the 95% confidence interval of a proportion using the plus-four method:

Proportion of respondents who are former smokers

110,060/426,000 = 0.2584

n= 426,000 and x= 110,060

Where n, as the total number of respondents and x, as the number of respondents who started they are former smokers.

Adding 4 to the n gives; n~= 426,000 + 4 = 426,004

Adding 2 to the x gives; x~ = 110,060 + 2 =110,062

Therefore, p = x~/n~ or 110,062/426,004 =0.2584

And q which is 1 – p or 1 -.2584 = .7416

Now, these numbers are plugged into the following equation to get your confidence intervals:

SEp = √ ((p*q)/n)

Calculating the parts in order gives;

1. Multiplying p x q: .1 x .7416 = 0.07416
2. Dividing the product of this by 426,004 gives; 0.07416/ 426,004 = 0.000000174
3. Obtaining the square root of the answer from step 2 gives; √.0.000000174= 0.0004172
4. Multiplying this by 1.96 to get the Sep gives; 1.96 x 0.0004172 = 0.0008178
5. Adding and subtracting the Sep to/from 0.2584 gives the confidence interval associated with the proportion: 0.2584 – 0.0008178 = 0.2576 and 0.2584 + 0.0008178 = 0.2592. Therefore, the confidence interval for the proportion of respondents who stated they are former smokers is (.2584, .2592) (Dupont, 2002).