Answer:
a) For this case and using the empirical rule we can find the limits in order to have 9% of the values:
[tex] \mu -2\sigma = 55 -2*6 =43[/tex]
[tex] \mu +2\sigma = 55 +2*6 =67[/tex]
95% of the widget weights lie between 43 and 67
b) For this case we know that 37 is 3 deviations above the mean and 67 2 deviations above the mean since within 3 deviation we have 99.7% of the data then the % below 37 would be (100-99.7)/2 = 0.15% and the percentage above 67 two deviations above the mean would be (100-95)/2 =2.5% and then we can find the percentage between 37 and 67 like this:
[tex] 100 -0.15-2.5 = 97.85[/tex]
c) We want to find the percentage above 49 and this value is 1 deviation below the mean so then this percentage would be (100-68)/2 = 16%
Step-by-step explanation:
For this case our random variable of interest for the weights is bell shaped and we know the following parameters.
[tex]\mu = 55, \sigma =6[/tex]
We can see the illustration of the curve in the figure attached. We need to remember that from the empirical rule we have 68% of the values within one deviation from the mean, 95% of the data within 2 deviations and 99.7% of the values within 3 deviations from the mean.
Part a
For this case and using the empirical rule we can find the limits in order to have 9% of the values:
[tex] \mu -2\sigma = 55 -2*6 =43[/tex]
[tex] \mu +2\sigma = 55 +2*6 =67[/tex]
95% of the widget weights lie between 43 and 67
Part b
For this case we know that 37 is 3 deviations above the mean and 67 2 deviations above the mean since within 3 deviation we have 99.7% of the data then the % below 37 would be (100-99.7)/2 = 0.15% and the percentage above 67 two deviations above the mean would be (100-95)/2 =2.5% and then we can find the percentage between 37 and 67 like this:
[tex] 100 -0.15-2.5 = 97.85[/tex]
Part c
We want to find the percentage above 49 and this value is 1 deviation below the mean so then this percentage would be (100-68)/2 = 16%
