Answer to Question #134739 in Statistics and Probability for Margaret

Question #134739
The number of times the AI algorithm is successful at detecting fake news is normally distributed with a sample mean of 900 and the sample standard deviation of 30. Assume a sample size of 100 was used.
What is the 95% confidence interval (CI) estimate for the population mean?
1. ( 894.1002 ; 905.8800)
2. (894.0474 ; 905.9526)
3. (852.8959 ; 947.1041)
4. (869.9665 ; 930.0335)
5. (895.0194 ; 904.9806)
1
Expert's answer
2020-09-27T17:55:18-0400



At the 95% confidence interval, the number of times the algorithm succeeds is between 2.5% and 97.5%


xˉ=900\bar x = 900 , σ=30\sigma=30 and n=100n=100


On the standard normal distribution;



Z score at 2.5%=1.96Z\ score\ at\ 2.5\% = -1.96Z score at 97.5%=1.96Z\ score\ at\ 97.5\% = 1.96


Z score=xxˉσnZ\ score =\frac{x - \bar x}{\frac{\sigma}{\sqrt{n}}}

Therefore:


x=σnZ score+xˉx =\frac{\sigma}{\sqrt{n}} * Z\ score +\bar x

The values of xx at the given Z score valueZ\ score \ value form the boundaries for the confidence interval


The lower boundary:



xlower=30100(1.96)+900=894.1200x_{lower}=\frac{30}{\sqrt{100}} * (- 1.96) + 900 =894.1200

xupper=301001.96+900=905.8800x_{upper}=\frac{30}{\sqrt{100}} * 1.96 + 900 = 905.8800

answer: the confidence interval is

[894.1200, 905.8800][894.1200,\ 905.8800]


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