The weight of a small Starbucks coffee is a normally distributed random variable
ID: 3046024 • Letter: T
Question
The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 365 grams and a standard deviation of 14 grams. Find the weight that corresponds to each event. (Use Excel or Appendix C to calculate the z-value. Round your final answers to 2 decimal places.)
The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 365 grams and a standard deviation of 14 grams. Find the weight that corresponds to each event. (Use Excel or Appendix C to calculate the z-value. Round your final answers to 2 decimal places.)
Explanation / Answer
Given that
X ~ N( mu = 365, sigma2 = 142)
a) Highest 10 percent
i.e P(a < X < b) = 0.10
P( ( a-mu)/sigma < (x-mu)/sigma < ( b-mu) / sigma) = 0.10
P( (a-365)/14 < z < (b-365)/14) = 0.10 where z ~ N(0,1)
From Normal probability table
P( 1.28 < z < 3) = 0.10
hence (a-365)/14 =1.28 and (b-365) /14 =3
a = 365 +1.28*14 and b = 365 + 3*14
a =382.92 and b= 407
b) Middle 50 percent
i.e P(a < X < b) =0.50
P( ( a-mu)/sigma < (x-mu)/sigma < ( b-mu) / sigma) = 0.50
P( (a-365)/14 < z < (b-365)/14) = 0.50 where z ~ N(0,1)
From Normal probability table
P( -0.68 < z < 0.68) = 0.50
hence (a-365)/14 =-0.68 and (b-365) /14 =0.68
a = 365 - 0.68*14 and b = 365 + 0.68*14
a =355.48 and b= 374.52
c) Highest 80 percent
i.e P(a < X < b) =0.80
P( ( a-mu)/sigma < (x-mu)/sigma < ( b-mu) / sigma) = 0.80
P( (a-365)/14 < z < (b-365)/14) = 0.80 where z ~ N(0,1)
From Normal probability table
P(-0.84 < z <3) = 0.80
hence (a-365)/14 = -0.84 and (b-365) /14 =3
a = 365 - 0.84*14 and b = 365 + 3 *14
a = 353.24 and b = 407
d) Lowest 10 percent
i.e P(a < X < b) =10
P( ( a-mu)/sigma < (x-mu)/sigma < ( b-mu) / sigma) = 10
P( (a-365)/14 < z < (b-365)/14) = 0.50 where z ~ N(0,1)
From Normal probability table
P(-3 < z < -1.28) = 0.10
hence (a-365)/14 =-3 and (b-365) /14 =-1.28
a = 365 - 3*14 and b = 365 -1.28*14
a = 323 and b= 347.08
a Highest 10 percent 382.92 to 407 b Middle 50 percent 355.48 to 374.52 c Highest 80 percent 353.24 to 407 d Lowest 10 percent 323 to 347.08Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.