Question

in a certain microwave oven on the high power setting, the time it takes a randomly chosen kernel of popcorn to pop is normally distributed with a mean of 140 seconds and a standard deviation of 25 seconds.
A) what percentage of the kernels will fail to pop if the popcorn is cooked for 2 minutes?
B) what percentage of the kernels will fail to pop if the popcorn is cooked for 3 minutes?
C) if you wanted 95% of the kernels to pop, what time would you allow?
D) if you wanted 99% of kernels to pop, what time would you allow?

Answer 1

Answer:

A) The percentage of the kernels will fail to pop if the popcorn is cooked for 2 minutes is 78.8%

B) The percentage of the kernels will fail to pop if the popcorn is cooked for 3 minutes is 5.48%

C) For 95% of the kernels to pop, the time to be allowed is 3 minutes and 11.13 seconds

D) For 99% of the kernels to pop, the time to be allowed is 3 minutes and 18.15 seconds

Step-by-step explanation:

A) The percentage of the kernels will fail to pop if the popcorn is cooked for 2 minutes

2 minutes = 120 seconds

z(120) = (120 - 140) / 25

= -20 / 25

= - 0.8

p(z > - 0.8) = 1 - p ( 2 < - 0.8)

= 1 - 0.2119

= 0.7881

= 78.8%

B) The percentage of the kernels will fail to pop if the popcorn is cooked for 3 minutes

3 minutes = 180 seconds

z(120) = (180 - 140) / 25

= 40 / 25

= 1.6

p(z > 1.6) = 1 - p ( 2 < 0.9452)

= 0.0548

= 5.48%

C) For 95% of the kernels to pop,  the time to be allowed is

The Z value that gives an area of 0.05 for the right tail > 1.645

From the formula ,

2 (x) = (x -μ ) / r

1.645 = (x - 140) / 25

x - 140 = 1.645 x 25

x - 140 = 41.12

x = 41.13 + 140

x = 181.13

Thus, For 95% of the kernels to pop,  the time to be allowed is 3 minutes and 11.13 seconds

D) For 99% of the kernels to pop, the time to be allowed is

The Z value that gives an area of 0.05 for the right tail > 2.326

From the formula ,

2 (x) = (x -μ ) / r

2.326 = (x - 140) / 25

x - 140 = 2.326 x 25

x - 140 = 58.15

x = 58.15 + 140

x = 198.15

Thus, For 99% of the kernels to pop, the time to be allowed is 3 minutes and 18.15 seconds