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Confidence Intervals
Part 1: Confidence Intervals Basics
Describe how confidence intervals are calculated and used.
The Confidence Interval (CI) refers to the scale of values that the “estimate” is expected to fall if the
statistical test is iterated at a certain confidence level determined by the z-distribution critical value
(Bevans, 2020). Calculating CI mathematically is shown below:
CI for the mean:
∝
√
𝑛
Whereby: CI =Confidence Interval
= Population Mean
Z= Z-distribution critical value
α = Standard deviation of the population
n= population size.
CI for a proportion.
√ 𝑃
̂ ( 1 −𝑃
̂ )
𝑛
Other signs are the same from the mean but 𝑃
= The sample Proportion (sample size/Population size);
and n= sample size.
Part 2: Confidence Intervals at 90% and 95% Confidence Levels
Use the sample data set to perform the following calculations on the mean internet usage of a
country in 2017:
● Calculate sample mean, standard deviation, and standard error.
● Calculate confidence intervals at 90% and 95% confidence levels.
The sample mean = Sum of the sample dataset / Number of the dataset (n)
From the spreadsheet, the mean, and the standard deviation (SD) obtained are as follows:
Standard Error (SE) can be calculated by taking the standard deviation (SD) divided by the
sample size square root (n^1/2) (Tuovila, 2022).
Calculating the confidence Intervals (CI) of 90%:
To calculate the CI, we need to get the z-score of 90% (0.9 ) under a sample size of less than 30.
Z(0.9) = 1. 75
The margin of error (ME)= SE Z-score*
Then the confidence Interval: Mean value-ME to Mean value + ME
: 75.594 - 5.40725 or 7 5.594 + 5.
:70.18675 to 81.
Calculating the confidence Intervals (CI) of 95%:
To calculate the CI, we need to get the z-score of 9 5 % (0.9 5 ) under a sample size of less than 30.
Z(0.9 5 ) = 2.
The margin of error (ME)= SE Z-score*
Then the confidence Interval: Mean value-ME to Mean value + ME
: 75.594 – 6.6961875 or 7 5.594 + 6.
:68.898 to 82.
Use the sample data set to perform the following calculations on the proportion of countries with
widespread internet usage in 2017 (widespread internet usage is defined as 75% or more of a
population using the internet):
● Calculate sample mean, standard deviation, and standard error.
● Calculate confidence intervals at 90% and 95% confidence levels.
Z-score * SE = 1.96 * 0.125 = 0.
Then:
Confidence Intervals of 95% = p’ - (Z-scoreSE) to p’ +(z-score SE)
Part 3: Conclusions
Using your work with confidence intervals, prepare your conclusion. In your response, address the
following:
● Compare the difference in the confidence intervals from 90% to 95%.
● The author stated that the average internet usage for a country in 2015 was 60%. Determine
whether this claim is within the confidence interval. Determine if the statement is a
reasonable claim.
● Draw conclusions based on the confidence intervals.
● Provide an overall recommendation on the accuracy of the author’s statements.
Confidence interval for the mean illustration:
The results of the difference between the Mean Value (MV) and the Margin of Error (ME) of the
confidence level of 90% resulted in 70.18675 while the summation of MV and ME gave 81.00125.
While the difference between MV and ME of the confidence level of 95% resulted in 68.898% while the
summation of MV and ME resulted in 82.29%. Now comparing the results of the difference between
MV and ME of 90% and 95% Confidence levels (CL), we can see that 90% resulted in a higher value
compared to 95%. This illustrates that the higher the confidence level the smaller the Confidence
Interval with the mean CI difference. When comparing the results of the summation between MV and
ME of 90% and 95% CL, we can see that 95% resulted in a higher value compared to 90%. This
illustrates that the higher the confidence level the higher the CI with the mean CI summation.
The confidence intervals for the Proportion illustration:
The results of the difference between sample Proportion (P’ or 𝑃
) and the Margin of Error of the
confidence level of 90% was 35.7% while the summation of the p’ and ME gives 76.8%. Nevertheless,
the difference between ME and the sample Proportion to Confidence Level 95% was 31.75% and the
summation with the ME gives 80.75%. From these results we can see the higher Confidence Interval on
the ME difference is on the smaller interval or smaller Confidence Level of the Proportion. Meanwhile,
the higher Confidence Interval on the ME summation is on the higher Confidence Level of the
Proportion.
General illustration:
Another key comparison I have discovered is that the Confidence Intervals of the Confidence Level of
95% lie within the confidence Intervals of the Confidence Level of 90%. For instance, the proportion
mean of 95%, 31.75% to 80.75% lies within the values 35.7% and 76.8% of the Confidence Level 90%.
The same applies to the CI of the mean. The 68.898% to 82.29% of the confidence Level 95%b lies
within 70.18675% and 81.00125% of the Confidence Level 90%.
However, another comparison to on the range of numbers. From the explanation above, we can see
that the Confidence Intervals of 90% on the mean extends up to 81% while 95% extends to 82%. The
1% increase for the 95% illustrates that the higher the confidence level the higher the values of the
mean Confidence Interval. The same applies to the CI of the proportions, the Ci of the 90%n is up to
76.8% while 95% is up to 80.75%. This approximately 4% more CI, illustrates that the higher the
confidence level of the Proportion the higher the CI or the spread of the CI values.
The proof of the 60% Claim:
To determine if the claim that the average internet usage for a country in 2015 was 60%, we need to
note the confidence levels obtained. The confidence level for 90% was to lie between 70.18675% to
81.00125%, from this point we see that the claim of 60% does not fall within the 90% intervals thus
making the claim not reasonable. In addition, the confidence level for 95% has to lie between 68.898%
to 82.29%, from this point we see that the claim of 60% does not fall within the 95% intervals, thus
making the claim not reasonable too.
Conclusion:
References
Bevans, R. (2020, August 7). Understanding Confidence Intervals | Easy Examples & Formulas.
From www.scribbr.com: https://www.scribbr.com/statistics/confidence-
interval/#:~:text=about%20confidence%20intervals-
,What%20exactly%20is%20a%20confidence%20interval%3F,a%20certain%20level%
of%20confidence.
Open Stax. (n.d.). 8.3 A Confidence Interval for A Population Proportion. From openstax.org:
https://openstax.org/books/introductory-business-statistics/pages/8- 3 - a-confidence-
interval-for-a-population-
proportion#:~:text=To%20calculate%20the%20confidence%20interval,find%20p%E2%
0%B2%2C%20q%E2%80%B2.&text=p%E2%80%B2%20%3D%200.842%20is%20the%
20samp
Tuovila, A. (2022, June 28). Standard Error of the Mean vs. Standard Deviation: What's the
Difference? From www.investopedia.com:
https://www.investopedia.com/ask/answers/042415/what-difference-between-standard-
error-means-and-standard-
deviation.asp#:~:text=SEM%20is%20calculated%20simply%20by,variability%20of%20th
e%20sample%20means.
