The diameter of an electric cable is normally distributed, with a mean of 0.8 inch and a standard deviation of 0.01 inch. What is the probability that the diameter will exceed 0.82 inch? (Give the answer to four decimal places.)
0.82 - 0.8 = 0.02
0.02/0.01 = 2 standard deviations
You check a normal distribution table (unless you were provided with the value for 2 standard deviation already)
You get : Greater than 2 standard deviation = 1 - 0.9772 = 0.0228
The diameter of an electric cable is normally distributed, with a mean of 0.8 inch and a standard deviation of 0.01 inch. What is the probability that the diameter will exceed 0.81 inch
To solve this problem, we will use the standard normal distribution formula.
Step 1: Convert the given values to z-scores.
Let X be the diameter of the electric cable.
Z = (X - μ) / σ
where μ is the mean and σ is the standard deviation.
In this case, X = 0.82 inch, μ = 0.8 inch, and σ = 0.01 inch.
Z = (0.82 - 0.8) / 0.01
Z = 0.02 / 0.01
Z = 2
Step 2: Find the probability using the standard normal distribution table.
The standard normal distribution table provides the probability for a given z-score. We want to find the probability that the diameter will exceed 0.82 inch, which is equivalent to finding the probability of having a z-score greater than 2.
Looking up the z-score of 2 in the standard normal distribution table, we find that the probability is approximately 0.9772.
Step 3: Calculate the complement.
The probability of the diameter exceeding 0.82 inch is 1 minus the probability of having a z-score less than or equal to 2.
P(D > 0.82) = 1 - P(Z ≤ 2)
P(D > 0.82) = 1 - 0.9772
P(D > 0.82) ≈ 0.0228
Therefore, the probability that the diameter will exceed 0.82 inch is approximately 0.0228.
To find the probability that the diameter will exceed 0.82 inch, we need to calculate the area under the normal distribution curve to the right of 0.82 inch.
1. Standardize the value: To use the z-table, we need to standardize the value of 0.82 inch using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
z = (0.82 - 0.8) / 0.01 = 0.02 / 0.01 = 2
2. Look up the z-score: Use a standard normal distribution table (z-table) to find the area to the right of 2. In the z-table, the values of the z-score are listed on the left-hand side and the probabilities are listed on the top. Find the closest z-score (2.0 in this case) and read the corresponding probability from the table.
Looking up the value of z = 2 in the z-table, we find that the corresponding probability is approximately 0.9772.
3. Calculate the probability of exceeding 0.82 inch: Since we are interested in the probability that the diameter exceeds 0.82 inch (area to the right of 0.82), we subtract the probability from 1.
Probability = 1 - 0.9772 = 0.0228
Therefore, the probability that the diameter will exceed 0.82 inch is approximately 0.0228.