We can solve each part of the problem using the properties of the standard normal distribution (a normal distribution with a mean of 0 and standard deviation of 1).

a. Probability that the thread length is within 1.5 standard deviations of the mean:

The probability that a value is within 1.5 standard deviations of the mean corresponds to the area under the standard normal curve between $-1.5$ and $1.5$. Using the standard normal table or a calculator, we find:

$P(-1.5 \leq Z \leq 1.5)$

First, find the cumulative probability for $Z = 1.5$ from a standard normal distribution:

$P(Z \leq 1.5) = 0.9332$

Since the normal distribution is symmetric around the mean:

$P(Z \geq -1.5) = P(Z \leq 1.5) = 0.9332$

Thus, the probability of being within 1.5 standard deviations of the mean is:

$P(-1.5 \leq Z \leq 1.5) = 0.9332 - (1 - 0.9332) = 0.9332 - 0.0668 = 0.8664$

So, the probability is 0.8664, or 86.64%.

b. Probability that the thread length is farther than 2.5 standard deviations from the mean:

This is the probability that the value lies outside the range of $-2.5$ to $2.5$. The formula would be:

$P(Z < -2.5 \text{ or } Z > 2.5) = 1 - P(-2.5 \leq Z \leq 2.5)$

Find the cumulative probability for $Z = 2.5$:

$P(Z \leq 2.5) = 0.9938$

Thus, the probability of being within 2.5 standard deviations of the mean is:

$P(-2.5 \leq Z \leq 2.5) = 0.9938$

Therefore, the probability of being farther than 2.5 standard deviations from the mean is:

$P(Z < -2.5 \text{ or } Z > 2.5) = 1 - 0.9938 = 0.0062$

So, the probability is 0.0062, or 0.62%.

c. Probability that the thread length is between 1 and 2 standard deviations from the mean:

This is the probability that the value lies between $1$ and $2$ standard deviations. The formula is:

$P(1 \leq Z \leq 2)$

Find the cumulative probabilities for $Z = 1$ and $Z = 2$:

$P(Z \leq 1) = 0.8413$ $P(Z \leq 2) = 0.9772$

Thus, the probability of being between 1 and 2 standard deviations is:

$P(1 \leq Z \leq 2) = P(Z \leq 2) - P(Z \leq 1) = 0.9772 - 0.8413 = 0.1359$

So, the probability is 0.1359, or 13.59%.

Summary:

• a. Probability within 1.5 SDs: 0.8664 (86.64%)
• b. Probability farther than 2.5 SDs: 0.0062 (0.62%)
• c. Probability between 1 and 2 SDs: 0.1359 (13.59%)

Would you like more details on how I calculated these probabilities?

Here are 5 related questions:

1. What is the probability of being within 2 standard deviations of the mean?
2. How would the probabilities change if the standard deviation increased?
3. What is the probability that a bolt thread length is more than 3 SDs from the mean?
4. How do you calculate z-scores for normally distributed data?
5. How does the cumulative distribution function (CDF) relate to these calculations?

Tip: In a standard normal distribution, approximately 68% of data falls within 1 SD, 95% within 2 SDs, and 99.7% within 3 SDs of the mean (the empirical rule).