A3. A Retailing Entrepreneur sells sophisticated running shoes whose weights are assumed to be
normally distributed with a mean of 12 grammes and a standard deviation of 0.5 grammes.
(a) What is the probability that a shoe weighs more than 13 grammes? [3]
(b) What must the standard deviation of weight be in order for the company to state that
99.9% of its shoes are less than 13 grammes? [6]
(c) If the standard deviation remains at 0.5 grammes, what must the mean weight be in
order for the company to state that 99.9% of its shoes are less than 13 grammes? [5]
(a) Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability that relates to the Z score.
(b,c) Reverse process from proportion to Z score. Insert into above equation and solve.
To find the probabilities in this scenario, we need to use the normal distribution and its properties. Here's how we can go about solving each part of the question:
(a) What is the probability that a shoe weighs more than 13 grammes?
To find this probability, we need to calculate the area under the normal distribution curve to the right of 13 grams. We can use the z-score formula to convert the value 13 grams to its corresponding z-score.
The z-score formula is given by:
z = (x - μ) / σ
where:
z = z-score
x = value we are interested in (in this case, 13 grams)
μ = mean of the distribution (12 grams)
σ = standard deviation of the distribution (0.5 grams)
Substituting the values into the formula:
z = (13 - 12) / 0.5
z = 1 / 0.5
z = 2
With the obtained z-score of 2, we can now find the probability using a standard normal distribution table or a calculator. The area to the right of 13 grams (z = 2) represents the probability that a shoe weighs more than 13 grams.
(b) What must the standard deviation of weight be in order for the company to state that 99.9% of its shoes are less than 13 grams?
To find the required standard deviation, we need to determine the z-score corresponding to the desired probability.
To have 99.9% of shoes weigh less than 13 grams, we need to calculate the z-score that corresponds to a probability of 0.999 (1 - 0.999 = 0.001). We can use a standard normal distribution table or a calculator to find the z-score.
Once we have the z-score, we can rearrange the z-score formula to solve for the standard deviation.
(c) If the standard deviation remains at 0.5 grams, what must the mean weight be in order for the company to state that 99.9% of its shoes are less than 13 grams?
To determine the required mean weight, we need to calculate the z-score corresponding to the desired probability (99.9%). Similar to part (b), we'll use a standard normal distribution table or a calculator to find the corresponding z-score.
Once we have the z-score, we can rearrange the z-score formula to solve for the mean weight, μ, while keeping the standard deviation at 0.5 grams.