Final answer:
Using the provided standard deviation and sample mean, along with the z-score for a 90% confidence interval, the true mean breaking strength of the cables is estimated to be between 1681.3 pounds and 1718.7 pounds.
Step-by-step explanation:
To find a 90% confidence interval for the true mean breaking strength of all cables produced by a certain manufacturer, where the standard deviation is known to be 80 pounds, and based on a random sample of 50 cables with a mean breaking strength of 1700 pounds, we use the formula for a confidence interval for the mean:
Confidence Interval = μ ± (z* × (σ/√n))
Here, μ is the sample mean, z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.
First, we find the z-score for a 90% confidence level, which is approximately 1.645. Then, we calculate the margin of error:
Margin of Error = 1.645 × (80/√50) = 1.645 × (80/7.071) = 18.749 pounds (rounded to three decimal places)
The 90% confidence interval is thus:
1700 ± 18.749 = (1681.3, 1718.7) pounds
Therefore, we are 90% confident that the true mean breaking strength of the cables lies between 1681.3 pounds and 1718.7 pounds, when rounded to one decimal place.