Respuesta :
Answer:
Step-by-step explanation:
Since the amount of soft drink dispensed into a cup is normally distributed, we would apply the formula for normal distribution which is expressed as
z = (x - µ)/σ
Where
x = amount in ounce of soft drink dispensed into cup.
µ = mean amount
σ = standard deviation
From the information given,
µ = 7.6oz
σ = 0.4 oz
a) The probability that the machine will overflow an 8-ounce cup is expressed as
P(x > 8) = 1 - P(x ≤ 8)
For x = 8,
z = (8 - 7.6)/0.4 = 1
Looking at the normal distribution table, the probability corresponding to the z score is 0.84
P(x ≤ 8) = 1 - 0.84 = 0.16
b) P(x< 8) = 0.84
c) when the machine has just been loaded with 848 cups, the number of cups expected to overflow when served is
0.16 × 848 = 136 cups
Using the normal distribution, it is found that:
a) 0.1587 = 15.87% probability that the machine will overflow an 8-ounce cup.
b) 0.8413 = 84.13% probability that the machine will not overflow an 8-ounce cup.
c) 135 cups are expected to overflow when served.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
In this problem:
- Mean of 7.6 oz, thus [tex]\mu = 7.6[/tex].
- Standard deviation of 0.4 oz, thus [tex]\sigma = 0.4[/tex].
Item a:
The probability of an overflow is the probability of the machine having 8 oz or more, thus, it is 1 subtracted by the p-value of Z when X = 8.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{8 - 7.6}{0.4}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a p-value of 0.8413.
1 - 0.8413 = 0.1587.
0.1587 = 15.87% probability that the machine will overflow an 8-ounce cup.
Item b:
0.1587 probability of an overflow, so 1 - 0.1587 = 0.8413 = 84.13% probability that the machine will not overflow an 8-ounce cup.
Item c:
0.1587 out of 848, thus:
[tex]0.1587(848) = 135[/tex]
135 cups are expected to overflow when served.
A similar problem is given at https://brainly.com/question/12490452
