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
Z = (score-mean)/SD
Here is a start:
(i) Z = (62-55)/6.2 = ?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to that Z score.
Could i got the full answer im lost
I need B
To solve these probability questions, we will use the cumulative distribution function (CDF) of the normal distribution. The general formula for calculating probabilities using the normal distribution is:
P(a < X < b) = P(X < b) - P(X < a)
Where X is a random variable with a normal distribution, a and b are the lower and upper limits, and P() is the cumulative distribution function.
Let's calculate the probabilities for each question:
(i) More than 62 units:
To find the probability of more than 62 units, we want to calculate P(X > 62). We can subtract the probability of X being less than or equal to 62 from 1.
P(X > 62) = 1 - P(X ≤ 62)
To find the probability using the standard normal distribution, we need to standardize the value using the z-score formula: z = (X - mean) / standard deviation.
Let's calculate the z-score for 62:
z = (62 - 55) / 6.2 = 1.13
Using the z-score table or a statistics calculator, we can find that the probability of Z being less than or equal to 1.13 is approximately 0.8708.
P(X > 62) = 1 - 0.8708 = 0.1292
So, the probability of observing more than 62 units of hand sanitizer is 0.1292 or 12.92%.
(ii) Less than 50 units:
To find the probability of less than 50 units, we want to calculate P(X < 50).
Let's calculate the z-score for 50:
z = (50 - 55) / 6.2 = -0.81
Using the z-score table or a statistics calculator, we can find that the probability of Z being less than or equal to -0.81 is approximately 0.2090.
P(X < 50) = 0.2090
So, the probability of observing less than 50 units of hand sanitizer is 0.2090 or 20.90%.
(iii) Less than 47 units:
Using the same process as in part (ii), we calculate the z-score for 47:
z = (47 - 55) / 6.2 = -1.29
Using the z-score table or a statistics calculator, we can find that the probability of Z being less than or equal to -1.29 is approximately 0.0985.
P(X < 47) = 0.0985
So, the probability of observing less than 47 units of hand sanitizer is 0.0985 or 9.85%.
(iiii) Between 45 and 66 units:
To find the probability of between 45 and 66 units, we want to calculate P(45 < X < 66) = P(X < 66) - P(X < 45).
Let's calculate the z-scores for 66 and 45:
For 66: z = (66 - 55) / 6.2 = 1.77
For 45: z = (45 - 55) / 6.2 = -1.61
Using the z-score table or a statistics calculator, we can find that the probability of Z being less than or equal to 1.77 is approximately 0.9616, and the probability of Z being less than or equal to -1.61 is approximately 0.0537.
P(45 < X < 66) = 0.9616 - 0.0537 = 0.9079
So, the probability of observing between 45 and 66 units of hand sanitizer is 0.9079 or 90.79%.
For part B) where 75% of days have a demand below a specific number of units, we need to calculate the value of X for which the cumulative distribution function equals 0.75.
Let's call this value a.
P(X < a) = 0.75
Using the z-score table or a statistics calculator, we can find that the z-score corresponding to a cumulative probability of 0.75 is approximately 0.674.
Using the z-score formula, we can solve for a:
0.674 = (a - 55) / 6.2
Simplifying the equation, we get:
a - 55 = 0.674 * 6.2
a - 55 = 4.169
a = 59.169
So, on 75% of days, the demand lies below approximately 59.169 units of hand sanitizer.