A) a retailor observes that the daily demand for hand sanitizer is normally distributed with a mean of 55 units and be a standard deviation of 6.2 units. calculate the probability
(i) more than 62 units
(ii) less than 50 unite
(iii) less than 47 units
(iiii) between 45 and 66 units
B) Sales figure show that on 75% of days demand lies below a specific number of units of hand sanitizers calculate this number of units
Probabilities table: i.imgur.com/xhxz3cc.png
To calculate the probabilities given the mean and standard deviation of a normal distribution, we can use the Z-score formula. The Z-score measures the number of standard deviations a value is from the mean.
The formula to calculate the Z-score is:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Value
μ = Mean
σ = Standard Deviation
To find the probabilities, we need to convert the given values into Z-scores and then refer to the probabilities table you provided.
(i) To calculate the probability of more than 62 units:
First, calculate the Z-score:
Z = (62 - 55) / 6.2
Z ≈ 1.129
Looking up the Z-score in the table, find the probability (P) corresponding to the Z-score. From the table, P(Z < 1.13) ≈ 0.8708.
However, we are interested in the probability of more than 62 units, which is 1 minus the probability we just found.
P(X > 62) = 1 - P(Z < 1.13)
P(X > 62) ≈ 1 - 0.8708
P(X > 62) ≈ 0.1292
(ii) To calculate the probability of less than 50 units:
Calculate the Z-score:
Z = (50 - 55) / 6.2
Z ≈ -0.806
From the table, find the probability (P) corresponding to the Z-score. P(Z < -0.81) ≈ 0.2090.
P(X < 50) ≈ 0.2090
(iii) To calculate the probability of less than 47 units:
Calculate the Z-score:
Z = (47 - 55) / 6.2
Z ≈ -1.29
From the table, find the probability (P) corresponding to the Z-score. P(Z < -1.29) ≈ 0.0985.
P(X < 47) ≈ 0.0985
(iiii) To calculate the probability between 45 and 66 units:
Calculate the Z-scores for both values:
Z1 = (45 - 55) / 6.2
Z1 ≈ -1.61
Z2 = (66 - 55) / 6.2
Z2 ≈ 1.77
From the table, find the probability (P) corresponding to each Z-score:
P(Z < -1.61) ≈ 0.0522
P(Z < 1.77) ≈ 0.9616
Now, calculate the probability between the two Z-scores:
P(-1.61 < Z < 1.77) ≈ P(Z < 1.77) - P(Z < -1.61)
P(-1.61 < Z < 1.77) ≈ 0.9616 - 0.0522
P(-1.61 < Z < 1.77) ≈ 0.9094
B)To calculate the number of units below which 75% of the days fall, we need to find the Z-score corresponding to a cumulative probability of 0.75.
From the probabilities table, find the Z-score that is closest to 0.75. In this case, the closest value is 0.7486 at the intersection of 0.7 and 0.04.
Looking at the Z-score value in the same row, it is approximately 0.6745.
Next, we can use the Z-score formula to find the corresponding value (X) of hand sanitizers:
Z = (X - μ) / σ
Rearranging the formula:
X = Z * σ + μ
X = 0.6745 * 6.2 + 55
X ≈ 59.125
Therefore, the number of units below which 75% of the days fall is approximately 59.125 units of hand sanitizers.