Question

Assuming the length of nails in a large box of 3 -inch nails is normally distributed with mean of 3 inches with standard deviation 0.02 Use the Cumulative Z-Score Table to answer the following questions. - What is the probability that a randomly selected nail from the box is longer than 3.007 inches? Write your answers in decimal form using 4 decimal places. - What is the probability that a randomly selected nail from the box is between 2.994 and 2.996 inches? Write your answers in decimal form using 4 decimal places. - If the manufacturer intends to discard the bottom 2.87% and the top 4.46% of nails, in terms of their lengths, what would be the cut-off lengths for nails to be discarded? Round your answers to two decimal places. Hint Low-end: inches High-end: inches

Answer 1

To find the probability and cut-off lengths for nails, we need to use the standard normal distribution and the Z-score. The probability of a longer nail is 0.6368, the probability of a nail between 2.994 and 2.996 inches is 0.0386, and the cut-off lengths for nails to be discarded are 2.96 inches and 3.035 inches.

Step-by-step explanation:

To answer these questions, we need to use the standard normal distribution and the Z-score. The Z-score measures the number of standard deviations an individual value is from the mean.

Question 1:

To find the probability that a randomly selected nail from the box is longer than 3.007 inches, we need to calculate the Z-score for 3.007 using the formula: Z = (X - μ) / σ.

Z = (3.007 - 3) / 0.02 = 0.35.

Using the Cumulative Z-Score Table, we can find that the probability corresponding to a Z-score of 0.35 is approximately 0.6368. Therefore, the probability that a randomly selected nail from the box is longer than 3.007 inches is 0.6368.

Question 2:

To find the probability that a randomly selected nail from the box is between 2.994 and 2.996 inches, we need to calculate the Z-scores for both values.

Z for 2.994 = (2.994 - 3) / 0.02 = -0.3.

Z for 2.996 = (2.996 - 3) / 0.02 = -0.2.

Using the Cumulative Z-Score Table, we can find that the probability corresponding to a Z-score of -0.2 is approximately 0.4207 and the probability corresponding to a Z-score of -0.3 is approximately 0.3821.

Therefore, the probability that a randomly selected nail from the box is between 2.994 and 2.996 inches is 0.4207 - 0.3821 = 0.0386.

Question 3:

To find the cut-off lengths for nails to be discarded, we need to calculate the Z-scores corresponding to the given percentages.

Z for the bottom 2.87% = Z for the area to the left of 0.0287 = -1.92 (approximately).

Z for the top 4.46% = Z for the area to the left of 0.9544 (1 - 0.0446) = 1.76 (approximately).

Using the Z-score formula: Z = (X - μ) / σ, we can solve for X.

For the bottom cut-off length: -1.92 = (X - 3) / 0.02. Solving for X, we find X = 2.96 (rounded to two decimal places).

For the top cut-off length: 1.76 = (X - 3) / 0.02. Solving for X, we find X = 3.035 (rounded to two decimal places).

Therefore, the cut-off lengths for nails to be discarded are 2.96 inches and 3.035 inches.