Answer:
a) P(X < 40) = 0.5
b) P(38 < X < 42) = 0.69143
c) P(X > 35) = 0.89435
d) P(X > 51) = 0.0029798
e) Find a meal cost which happen to be cheaper than 10% of the meals based on a given data. = $34.872
f) Find a meal cost which happen to be in the 5% of the most expensive based on a given data = $33.42
g) Find a meal costs which happen to be 40% symmetrically around the mean value based on a given data. = $38.988
Explanation:
Z score formula = Z score = x - μ/σ
Mean = $40
Standard deviation = $4
a) P(X < 40)
Z score = x - μ/σ
= 40 - 40/4
= 0
Determining the Probability value from Z-Table:
P(X < 40) = 0.5
b) P(38 < X < 42)
For X = 38
Z score = x - μ/σ
= 38 - 40/4
= -0.5
Determining the Probability value from Z-Table:
P(X = 38) = 0.30854
For X = 42
Z score = x - μ/σ
= 42 - 40/4
= 4
Determining the Probability value from Z-Table:
P(x = 42) = 0.99997
Hence, P(38 < X < 42)
P(X = 42) - P(X = 38)
0.99997 - 0.30854
= 0.69143
c) P(X > 35)
Z score = x - μ/σ
= 35 - 40/4
= -1.25
Determining the Probability value from Z-Table:
P( X < 35) = 0.10565
P( X > 35) = 1 - P(X < 35)
1 - 0.10565
= 0.89435
d) P(X > 51)
Z score = x - μ/σ
= 51 - 40/4
= 2.75
Determining the Probability value from Z-Table:
P( X < 51) = 0.99702
P(X > 51) = 1 - P(X < 51)
= 1 - 0.99702
= 0.0029798
e) Find a meal cost which happen to be cheaper than 10% of the meals based on a given data.
z score for 10th percentile = -1.282
Z score formula = Z score = x - μ/σ
Mean = $40
-1.282 = x - 40/4
-1.282 × 4 = x - 40
-5.128 + 40 = x
$34.872
The meal cost which happen to be cheaper than 10% of the meals based on a given data is $34.872
f) Find a meal cost which happen to be in the 5% of the most expensive based on a given data
Z score for 5th percentile = -1.645
Z score formula = Z score = x - μ/σ
Mean = $40
-1.645 = x - 40/4
-1.645 × 4 = x - 40
-6.58 + 40 = x
$33.42
The meal cost which happen to be in the 5% of the most expensive based on a given data is $33.42
g) Find a meal costs which happen to be 40% symmetrically around the mean value based on a given data.
Z score for 40th percentile = -0.253
Z score formula = Z score = x - μ/σ
Mean = $40
-0.253 = x - 40/4
-0.253 × 4 = x - 40
-1.012 + 40 = x
$38.988