Answer to Question #134739 in Statistics and Probability for Margaret

Question

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)

Answer 1

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

$\bar x = 900$ , $\sigma=30$ and $n=100$

On the standard normal distribution;

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

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

Therefore:

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

The values of $x$ at the given $Z\ score \ value$ form the boundaries for the confidence interval

The lower boundary:

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

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

answer: the confidence interval is

$$[894.1200,\ 905.8800]$$