Respuesta :
We can solve this question by using the fact that a normal distribution is symmetric and using the empirical rules. The empirical rules are
• 68% of the data falls within one standard deviation of the mean.
• 95% of the data falls within two standard deviations of the mean.
• 99.7% of the data falls within three standard deviations of the mean.
In our problem, the mean is 28 and the standard deviation is equal to 4. Our interval is 24(one standard deviation below the mean) and 36(2 standard deviations above the mean)
[tex](24,36)=(28-4,28+8)=(28-\sigma,28+2\sigma)[/tex]Our area is the area that falls within one standard deviation of the mean plus the area between one standard deviation and two on the positive side. The first area is 68% of the data, and the second area we can calculate using the fact that a bell curve is symmetric. Since 68% of data falls within one standard deviation and 95% of the data falls within two standard deviations of the mean, the difference between them represents the data that falls between one standard deviation and two standard deviation.
[tex]95-68=27[/tex]Since the distribution is symmetric, the data that falls between one standard deviation and two standard deviation on the positive side is half of this value.
[tex]\frac{27}{2}=13.5[/tex]Now, we just add this value to 68% and we're going to have our answer.
[tex]68+13.5=81.5[/tex]The percentage of employees worked between 24 and 36 hours is 81.5%.
