The lengths of what are considered "1 inch bolts are found to be normally distrubted with a mean of 1.001 inches and a standard deviation of 0.002 inches. If a bolt measures more than 2 standard deviations away from the mean, it is rejected for not meeting factory tolerances.
a. what are the minimum and maximum lengths required for a bolt to meet factory tolerances?
b. what percentage of bolts will be rejected?
You need to memorize this, pronto.
In addition to being more understandable than the variance as a measure of the amount of variation in the data, the standard deviation summarizes how close observations are to the mean in a very nice way. Many variables in biology fit the normal probability distribution fairly well. If a variable fits the normal distribution, 68.3 percent (or roughly two-thirds) of the values are within one standard deviation of the mean, 95.4 percent are within two standard deviations of the mean, and 99.7 (or almost all) are within 3 standard deviations of the mean.
within 1 standard deviation: 68.3 %
within 2 standard deviations: 95.3 %
within 3 standard deviations: 99.7 %
a. To determine the minimum and maximum lengths required for a bolt to meet factory tolerances, we need to calculate the cut-off points which are 2 standard deviations away from the mean.
The mean length of the 1-inch bolts is 1.001 inches, and the standard deviation is 0.002 inches.
Lower Cut-off Point:
Mean - (2 * Standard Deviation) = 1.001 - (2 * 0.002) = 0.997 inches
Upper Cut-off Point:
Mean + (2 * Standard Deviation) = 1.001 + (2 * 0.002) = 1.005 inches
Therefore, any bolt length below 0.997 inches or above 1.005 inches would be rejected for not meeting factory tolerances.
b. To calculate the percentage of bolts that will be rejected, we need to find the area under the normal curve outside the acceptable range.
First, we find the Z-scores for the lower and upper cut-off points:
Lower Z-score = (Lower Cut-off Point - Mean) / Standard Deviation = (0.997 - 1.001) / 0.002 = -2
Upper Z-score = (Upper Cut-off Point - Mean) / Standard Deviation = (1.005 - 1.001) / 0.002 = 2
Using a standard normal distribution table or a Z-score calculator, we can find the proportion of the distribution outside the range of -2 to 2.
The proportion of the distribution outside this range represents the percentage of bolts that will be rejected.
Looking up the Z-score of -2 in the standard normal distribution table gives a cumulative probability of approximately 0.0228.
Since the normal distribution is symmetric, the upper tail also has a cumulative probability of approximately 0.0228.
Therefore, the percentage of bolts that will be rejected is:
(0.0228 + 0.0228) * 100 = 4.56%
Approximately 4.56% of the bolts will be rejected for not meeting factory tolerances.
a. To find the minimum and maximum lengths required for a bolt to meet factory tolerances, we need to calculate the values that are two standard deviations away from the mean.
Step 1: Calculate the value of two standard deviations:
Multiply the standard deviation by 2, and then add it to the mean to get the maximum acceptable length:
Maximum length = mean + (2 * standard deviation)
Step 2: Calculate the minimum acceptable length:
Subtract two standard deviations from the mean to get the minimum acceptable length:
Minimum length = mean - (2 * standard deviation)
Let's calculate the values using the given information:
Mean = 1.001 inches
Standard Deviation = 0.002 inches
Maximum length = 1.001 + (2 * 0.002)
= 1.001 + 0.004
= 1.005 inches
Minimum length = 1.001 - (2 * 0.002)
= 1.001 - 0.004
= 0.997 inches
Therefore, the minimum length required for a bolt to meet factory tolerances is 0.997 inches, and the maximum length required is 1.005 inches.
b. To calculate the percentage of bolts that will be rejected, we need to determine the proportion of bolts that fall outside the acceptable range.
Step 1: Calculate the z-score:
Subtract the mean from the maximum acceptable length and divide by the standard deviation:
z-score = (maximum length - mean) / standard deviation
Step 2: Use a standard normal distribution table or a calculator to find the proportion of observations beyond the calculated z-score. As the z-score measures the number of standard deviations from the mean, we can use the table to find the cumulative probability. For a value beyond 2 standard deviations, we need to find the probability of values greater than the z-score.
The proportion of bolts that will be rejected is given by (1 - cumulative probability).
Let's calculate it using the information provided:
Maximum length = 1.005 inches
Mean = 1.001 inches
Standard Deviation = 0.002 inches
z-score = (maximum length - mean) / standard deviation
= (1.005 - 1.001) / 0.002
= 4
Using a standard normal distribution table or a calculator, we find that the cumulative probability for a z-score of 4 is approximately 0.99997.
Therefore, the proportion of bolts that will be rejected is (1 - 0.99997) = 0.00003, which is equivalent to 0.003% or 0.003 out of every 1000 bolts.