Question

Consider a sample with a mean of 60 and a standard deviation of 4. Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges (to the nearest whole number).

40 to 80, at least %

45 to 75, at least %

52 to 68, at least %

47 to 73, at least %

44 to 76, at least %

Answer 1

as we know that from Chebychev: k std deviation away from mean data values have

percentage of the data =(1-1/k²)*100

1)

for k score =(X-mean)/std deviation ; here k=(80-60)/4=5

hence  percentage of the data within this range =(1-1/5²)*100=96%

2)

k score=(75-60)/4 =3.75

hence  percentage of the data within this range =(1-1/3.75²)*100=92.89%

3)

k score=(68-60)/4 =2

hence  percentage of the data within this range =(1-1/2²)*100=75.0%

4)

k score=(73-60)/4 =3.25

hence  percentage of the data within this range =(1-1/3.25²)*100=90.53%

5)

k score=(76-60)/4 =4

hence  percentage of the data within this range =(1-1/4²)*100=93.75%