the average number of gallons of lemonade consumed by the football team during a game is 20, with a standard deviation of 3 gallons. assume the variable is normally distributed. when a game is played, find the probability of using
a) Between 20 and 25 gallons
b) less than 19 gallons
c) More than 21 gallons
d) between 26 and 28 gallons
could u post the solution for this plz
To solve these probability questions, we'll need to calculate the z-scores and use the standard normal distribution table.
First, let's define the mean and standard deviation of the lemonade consumption:
Mean (μ) = 20 gallons
Standard Deviation (σ) = 3 gallons
a) Probability between 20 and 25 gallons:
To find this probability, we'll calculate the area under the curve between the z-scores of 20 and 25.
Step 1: Find the z-score for the lower limit (20 gallons):
z1 = (X - μ) / σ
z1 = (20 - 20) / 3
z1 = 0
Step 2: Find the z-score for the upper limit (25 gallons):
z2 = (X - μ) / σ
z2 = (25 - 20) / 3
z2 = 1.67
Step 3: Look up the z-scores in the standard normal distribution table to find the areas:
P(20 < X < 25) = P(0 < Z < 1.67)
Using the table, we find that the area to the left of 1.67 is approximately 0.9525.
So, P(0 < Z < 1.67) = 0.9525
b) Probability less than 19 gallons:
To find this probability, we'll calculate the area under the curve up to the z-score of 19.
Step 1: Find the z-score for the given value (19 gallons):
z = (X - μ) / σ
z = (19 - 20) / 3
z = -0.33
Step 2: Look up the z-score in the standard normal distribution table to find the area:
P(X < 19) = P(Z < -0.33)
Using the table, we find that the area to the left of -0.33 is approximately 0.3707.
So, P(Z < -0.33) = 0.3707
c) Probability more than 21 gallons:
To find this probability, we'll calculate the area under the curve from the z-score of 21 to positive infinity (∞).
Step 1: Find the z-score for the given value (21 gallons):
z = (X - μ) / σ
z = (21 - 20) / 3
z = 0.33
Step 2: Look up the z-score in the standard normal distribution table to find the area:
P(X > 21) = P(Z > 0.33)
Using the table, we find that the area to the left of 0.33 is approximately 0.6293.
So, P(Z > 0.33) = 0.3707
d) Probability between 26 and 28 gallons:
To find this probability, we'll calculate the area under the curve between the z-scores of 26 and 28.
Step 1: Find the z-score for the lower limit (26 gallons):
z1 = (X - μ) / σ
z1 = (26 - 20) / 3
z1 = 2
Step 2: Find the z-score for the upper limit (28 gallons):
z2 = (X - μ) / σ
z2 = (28 - 20) / 3
z2 = 2.67
Step 3: Look up the z-scores in the standard normal distribution table to find the areas:
P(26 < X < 28) = P(2 < Z < 2.67)
Using the table, we find that the area to the left of 2 is approximately 0.9772,
and the area to the left of 2.67 is approximately 0.9953.
So, P(2 < Z < 2.67) = 0.9953 - 0.9772 = 0.0181
Therefore:
a) P(20 < X < 25) = 0.9525
b) P(X < 19) = 0.3707
c) P(X > 21) = 0.6293
d) P(26 < X < 28) = 0.0181
To solve these probability questions, we can make use of the Z-score formula and the Z-table. The Z-score measures the number of standard deviations a data point is from the mean. We'll need to calculate the Z-scores for each scenario and use the Z-table to determine the corresponding probabilities.
a) Between 20 and 25 gallons:
To find the probability of consuming between 20 and 25 gallons, we need to calculate the Z-scores for both endpoints and then find the probability between those Z-scores.
Z1 = (X1 - μ) / σ
Z1 = (20 - 20) / 3 = 0
Z2 = (X2 - μ) / σ
Z2 = (25 - 20) / 3 = 1.67
Using the Z-table, we can find the probability corresponding to Z1 and Z2.
P(20 ≤ X ≤ 25) = P(0 ≤ Z ≤ 1.67) = P(Z ≤ 1.67) - P(Z ≤ 0)
You can find the corresponding probabilities from the Z-table. Subtracting the two probabilities will give us the desired probability.
b) Less than 19 gallons:
To find the probability of consuming less than 19 gallons, we need to calculate the Z-score for 19 and find the probability corresponding to it.
Z = (X - μ) / σ
Z = (19 - 20) / 3 = -0.33
Using the Z-table, we can find the probability corresponding to this Z-score.
P(X < 19) = P(Z < -0.33)
c) More than 21 gallons:
To find the probability of consuming more than 21 gallons, we need to calculate the Z-score for 21 and find the probability corresponding to it.
Z = (X - μ) / σ
Z = (21 - 20) / 3 = 0.33
Using the Z-table, we can find the probability associated with this Z-score.
P(X > 21) = 1 - P(Z ≤ 0.33)
d) Between 26 and 28 gallons:
To find the probability of consuming between 26 and 28 gallons, we follow the same procedure as in part (a).
Z1 = (X1 - μ) / σ
Z1 = (26 - 20) / 3 = 2
Z2 = (X2 - μ) / σ
Z2 = (28 - 20) / 3 = 2.67
Using the Z-table, we can find the probability corresponding to Z1 and Z2.
P(26 ≤ X ≤ 28) = P(2 ≤ Z ≤ 2.67) = P(Z ≤ 2.67) - P(Z ≤ 2)
Please note that the Z-table values may vary slightly depending on the specific table you're using, as it approximates the cumulative probability of the standard normal distribution.