a retailing entrepreneur sells sophisticated running shoes whose weights are normally distributed with mean of 12 grams and a standard deviation of 0,5 grams.
(a) what is the probability that a shoe weighs more than 13 grams
(b) what must be the standard deviation of weight be in order for the company to state that 99,9% of its shoes are less than 13 grams
(c) if the standard deviation remains at 0,5 grams, what must be the mean weight in order for the company to state that 99,9% of its shoes are less that 13 grams.
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(a) To find the probability that a shoe weighs more than 13 grams, we need to calculate the area under the normal distribution curve above the value of 13 grams.
To do this, we can use a standard normal distribution table or a statistical calculator.
Using a standard normal distribution table:
1. Convert the value of 13 grams to a z-score. The formula to calculate the z-score is:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
In this case, x = 13 grams, μ = 12 grams, and σ = 0.5 grams.
Plugging these values into the formula, we get:
z = (13 - 12) / 0.5 = 2
2. Look up the z-score of 2 in the standard normal distribution table. The area to the left of 2 is 0.9772.
3. Subtract the area from 1 to get the area to the right of 2. The probability that a shoe weighs more than 13 grams is:
P(X > 13) = 1 - 0.9772 = 0.0228
Therefore, the probability that a shoe weighs more than 13 grams is approximately 0.0228 or 2.28%.
(b) To find the standard deviation necessary for the company to state that 99.9% of its shoes are less than 13 grams, we need to find the z-score corresponding to the desired cumulative probability.
Using a standard normal distribution table:
1. Look for the cumulative probability of 0.999 at the table.
2. Find the corresponding z-score. In this case, the z-score is approximately 3.09.
3. Use the z-score formula to find the standard deviation:
z = (x - μ) / σ
Rearranging the formula, we get:
σ = (x - μ) / z
Plugging in the values, x = 13 grams, μ = 12 grams, and z = 3.09, we get:
σ = (13 - 12) / 3.09 ≈ 0.3247
Therefore, the standard deviation of the weight must be approximately 0.3247 grams for the company to state that 99.9% of its shoes are less than 13 grams.
(c) If the standard deviation remains at 0.5 grams, we need to find the mean weight necessary for the company to state that 99.9% of its shoes are less than 13 grams.
Using a standard normal distribution table:
1. Look for the cumulative probability of 0.999 at the table.
2. Find the corresponding z-score. In this case, the z-score is approximately 3.09.
3. Use the z-score formula to find the mean:
z = (x - μ) / σ
Rearranging the formula, we get:
μ = x - (σ * z)
Plugging in the values, x = 13 grams, σ = 0.5 grams, and z = 3.09, we get:
μ = 13 - (0.5 * 3.09) ≈ 11.4545
Therefore, the mean weight must be approximately 11.4545 grams for the company to state that 99.9% of its shoes are less than 13 grams, assuming the standard deviation remains at 0.5 grams.