# Calculating Confidence Intervals

Calculating Confidence Intervals

Calculating Confidence Intervals
Problem 4.1:
The 95% confidence intervals for the mean of the variable weight

One-Sample Statistics

N
Mean
Std. Deviation
Std. Error Mean

weight
53
1.5875E2
35.64888
4.89675

One-Sample Test

Test Value = 0

t
df
Sig. (2-tailed)
Mean Difference
95% Confidence Interval of the Difference

Lower
Upper

weight
32.420
52
.000
158.75472
148.9287
168.5808

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

One-Sample Test

Test Value = 0

t
df
Sig. (2-tailed)
Mean Difference
90% Confidence Interval of the Difference

Lower
Upper

weight
32.420
52
.000
158.75472
150.5542
166.9553

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

One-Sample Test

Test Value = 0

t
df
Sig. (2-tailed)
Mean Difference
99% Confidence Interval of the Difference

Lower
Upper

weight
32.420
52
.000
158.75472
145.6621
171.8473

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:

Total number of respondents
Number of respondents who smoke everyday
Number of respondents who smoke some days
Number of respondents who are former smokers
Number of respondents who have never smoked

426,000
54,815
21,581
110,060
143,619

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

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;

Multiplying p x q: .1 x .7416 = 0.07416
Dividing the product of this by 426,004 gives; 0.07416/ 426,004 = 0.000000174
Obtaining the square root of the answer from step 2 gives; √.0.000000174= 0.0004172
Multiplying this by 1.96 to get the Sep gives; 1.96 x 0.0004172 = 0.0008178
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).

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