The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 248.5 and a standard deviation of 61.1. (all units are 1000 cells/mul.) using the empirical rule, find each approximate percentage below.a. what is the approximate percentage of women with platelet counts within 2 standard deviations of the mean, or between 126.3 and 370.7?b. what is the approximate percentage of women with platelet counts between 65.2 and 431.8?
Accepted Solution
A:
Answer:a. Approximately 95% women have platelet counts between 126.3 and 370.7.b. Approximately 99.7% women have platelet counts between 65.2 and 431.8.Step-by-step explanation: We have been given that the blood platelet counts of a group of women have a bell-shaped distribution with a mean of 248.5 and a standard deviation of 61.1.a. Empirical rule of normal distribution states: [tex]\approx68\%[/tex] of the data lies within 1 standard deviation of the mean.[tex]\approx95\%[/tex] of the data lies within 2 standard deviations of the mean.[tex]\approx99.7\%[/tex] of the data lies within 3 standard deviations of the mean.Now let us find z-score for our given values as z-score represents that a data point is how many standard deviation away from mean.[tex]z=\frac{x-\mu}{\sigma}[/tex][tex]z=\frac{126.3-248.5}{61.1}[/tex][tex]z=\frac{-122.2}{61.1}[/tex][tex]z=-2[/tex][tex]z=\frac{370.7-248.5}{61.1}[/tex][tex]z=\frac{122.2}{61.1}[/tex][tex]z=2[/tex]Since our given values are -2 and 2 standard deviations of mean, therefore, approximately 95% of women have platelet counts between 126.3 and 370.7.b. Let us find z-score for our given values as z-score represents that a data point is how many standard deviation away from mean.[tex]z=\frac{x-\mu}{\sigma}[/tex][tex]z=\frac{65.2-248.5}{61.1}[/tex][tex]z=\frac{-183.3}{61.1}[/tex][tex]z=-3[/tex] [tex]z=\frac{431.8-248.5}{61.1}[/tex][tex]z=\frac{183.3}{61.1}[/tex][tex]z=3[/tex]Since our given values are -3 and 3 standard deviations of mean, therefore, approximately 99.7% of women have platelet counts between 65.2 and 431.8.