Final answer:
To find the probability that a bolt width is between 945 and 955 mm, calculate the z-scores for both values, use a standard normal distribution table to find the corresponding probabilities, and subtract them to obtain the final answer, rounded to four decimal places.
Step-by-step explanation:
To determine the probability that a randomly chosen bolt of fabric has a width between 945 and 955 mm, given that the mean width is 951 mm with a standard deviation of 10 mm, we can use the properties of the normal distribution. First, we standardize the values by computing the z-scores for 945 mm and 955 mm.
To calculate the z-score, use the formula z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
- For 945 mm: z = (945 - 951) / 10 = -0.6
- For 955 mm: z = (955 - 951) / 10 = 0.4
Next, we consult the standard normal distribution table (or use a calculator) to find the probabilities corresponding to these z-scores. The probability of a z-score being between -0.6 and 0.4 reflects the probability of the bolt width being between 945 and 955 mm.
Let's denote these probabilities as P(z < 0.4) and P(z < -0.6). The probability we are looking for is P(z < 0.4) - P(z < -0.6). The standard normal distribution table provides these probabilities or they can be found using technology.
Once we have these probabilities, we subtract the lower probability from the higher probability to find the probability of a bolt width in that range. After finding the difference, we round the result to four decimal places as instructed.