Agroup of engineers developed a new design for a steel cable. they need to estimate the amount of weight the cable can hold. the weight limit will be reported on cable packaging. the engineers take a random sample of 41 cables and apply weights to each of them until they break. the 41 cables have a mean breaking weight of 772.3 lb. the standard deviation of the breaking weight for the sample is 15.3 lb. find the 95% confidence interval to estimate the mean breaking weight for this type cable. ( 11.74 , 12.86 ) your answer should be to 2 decimal places.

The confidence interval is a one of the estimates that can be determined from an observed data

The confidence interval is:

The given parameters are:

Start by calculating the standard deviation of the sample

Next, calculate the degree of freedom (df)

So, we have:

Calculate the significance level

The T value, when and us:

The endpoints of the confidence interval is calculated as:

So, we have:

Split

Hence, the confidence interval is:

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The 95% confidence interval for the mean breaking weight for this type cable is (767.47 lb, 777.13 lb).

Step-by-step explanation:

Our sample size is 41

The first step to solve this problem is finding our degrees of freedom, that is, the sample size subtracted by 1. So

Then, we need to subtract one by the confidence level and divide by 2. So:

Now, we need our answers from both steps above to find a value T in the t-distribution table. So, with 40 and 0.025 in the two-sided t-distribution table, we have

Now, we find the standard deviation of the sample. This is the division of the standard deviation by the square root of the sample size. So

Now, we multiply T and s

Then

The lower end of the confidence interval is the mean subtracted by M. So:

The upper end of the confidence interval is the mean added to M. So:

The 95% confidence interval for the mean breaking weight for this type cable is (767.47 lb, 777.13 lb).