Solution
- This is a binomial probability problem because we have multiple trials.
- The formula for calculating the Z-value is:
[tex]\begin{gathered} Z=\frac{X-\mu}{\sigma} \\ where, \\ \mu=\text{ The mean} \\ \sigma=\text{ The standard deviation} \\ X=\text{ The value we are testing} \end{gathered}[/tex]- This value of Z can be used to calculate the probability we need using a Z-score calculator or a Z-distribution table.
- Before we proceed, we need to find the mean and the standard deviation as follows:
[tex]\begin{gathered} \mu=np \\ n=\text{ Number of subjects} \\ p=\text{ The probability of success} \\ \\ \mu=\frac{6}{100}\times538 \\ \\ \mu=32.28 \\ \\ \sigma=\sqrt{np(1-p)} \\ \sigma=\sqrt{538\times\frac{6}{100}(1-\frac{6}{100})} \\ \\ \therefore\sigma=5.5085 \end{gathered}[/tex]- Now that we have both the mean and the standard deviation, we can proceed to find the value of the Z-score as follows:
[tex]\begin{gathered} Z=\frac{X-\mu}{\sigma} \\ \\ Z=\frac{28-32.28}{5.5085} \\ \\ \therefore Z=-0.78\text{ \lparen To 2 decimal places\rparen} \end{gathered}[/tex]- Now that we have the Z-score value, we can proceed to find the corresponding probability for values less than X = 28 sales using a Z-distribution table or a Z-score calculator.
- Using a Z-score calculator, we have:
- Since we are looking for the probability of having sales lower than 28, we have:
[tex]P(X
