Answer:
a) 0.10565
b) The two values = 4.61712 and
4.78288
c) 4.64088
Step-by-step explanation:
The z score formula we use to solve when given a random number of samples=
The formula for calculating a z-score is is z = (x-μ)/σ/√n
where x is the raw score
μ is the population mean
σ/√n is standard error
σ is the population standard deviation
n = random number of samples
a. What is the probability that the sample mean amount of juice will be at least 4.60 ounces?
z = (x-μ)/σ/√n
x = 4.60
μ = 4.70
σ = 0.40
n = 25
z = 4.60 - 4.70/0.40/√25
z = -0.10/0.40/5
z = -1.25
P-value from Z-Table:
P(x ≤ 4.60)
= P(x ≤ 4.60)
= P(z = -1.25)
= 0.10565
Therefore, the probability that the sample mean amount of juice will be at least 4.60 ounces is 0.10565
b. The probability is 70% that the sample mean amount of juice will be contained between what two values symmetrically distributed around the population mean?
The probability of 70% = 0.70
The z score for 0.70 = 1.036
This means, -1.036 and +1.036 gives a probability (0.70) or confidence level of 70%
We are to find x(raw score)
We solve using z score formula
For z = +1.036
z = (x-μ)/σ/√n
x = ?
μ = 4.70
σ = 0.40
n = 25
1.036 = x - 4.70/0.40/√25
1.036 = x - 4.70/0.08
Cross Multiply
1.036 × 0.08 = x - 4.70
0.08288 = x - 4.70
0.08288 + 4.70 = x
x = 4.78288
For z = -1.036
z = (x-μ)/σ/√n
x = ?
μ = 4.70
σ = 0.40
n = 25
-1.036 = x - 4.70/0.40/√25
-1.036 = x - 4.70/0.08
Cross Multiply
-1.036 × 0.08 = x - 4.70
-0.08288 = x - 4.70
-0.08288 + 4.70 = x
= 4.61712
Therefore, the two values = 4.61712 and
4.78288
c. The probability is 77% that the sample mean amount of juice will be greater than what value?
We are told to find the value that the sample mean is greater than
The z score for a P(x > z)
= P(x > 0.77)
= -0.739
Using z score formula, we find x first
z = (x-μ)/σ/√n
x = ?
μ = 4.70
σ = 0.40
n = 25
-0.739 = x - 4.70/0.40/√25
-0.739 = x - 4.70/0.08
Cross Multiply
-0.739 × 0.08 = x - 4.70
-0.05912 = x - 4.70
-0.05912 + 4.70 = x
x = 4.64088
Therefore, the value that the probability is 77% that the sample mean amount of juice will be greater than is 4.64088
