standard deviation

Question

The breaking strengths of cables produced by a certain manufacturer have a standard deviation of 108

 pounds. A random sample of 50

 newly manufactured cables has a mean breaking strength of 1700

 pounds. Based on this sample, find a 99%

 confidence interval for the true mean breaking strength of all cables produced by this manufacturer. Then give its lower limit and upper limit.

Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)

Lower limit:
Upper limit:

Answer 1

Hi Aubree,

We do not know whether this data is normally distributed, so we will use a t-confidence interval. The formula for that:

CI= x-bar +/- t*SE

Let's break that down:

x-bar=sample mean

t*=t critical value, obtained from t-table or statistical software

SE=standard error, which has its own formula as discussed below

For samples, we need to compute standard error. This is:

SE=s/sqrt(n) where:

s=standard deviation

n=sample size

Let's go ahead and calculate this for our problem

s=108

n=1500

SE=108/sqrt(1500)

SE=15.274

Now, looking back at our original equation for the confidence interval:

Ci=x-bar +/- t*SE,

we now have:

x-bar=1700

SE=15.274

We still need t*, the t-critical value. To compute this, we first need degrees of freedom. To calculate degrees of freedom, the formula is:

df=n-1

n=sample size

So:

n=50

df=n-1=49

Now, we need either a t-table or statistical software. If we use the t-table we have to look at the row closest to our degrees of freedom without going over and our alpha level. For a 99% confidence interval, assuming it is two-sided, we want alpha=0.005. This is (1-0.99)/2, so we control for both sides. You may want to memorize that alpha, since most confidence intervals in classroom practice--at least in my experience--are two-sided. You could also memorize 0.025 for 95 %, as this confidence level is also common.

Anyway, from t-table, closest df without going over is 40:

t^*₄₀=2.704

Finally, we have all values we need to compute the confidence interval:

CI=x-bar +/- t*SE

x-bar=1700

t*=2.704

SE=15.274

CI=1700 +/- (2.704-15.274)

Rounding to one decimal:

CI=1700 +/- 41.3

CI=(1658.7, 1741.3)

Lower Bound=1658.7

Upper Bound=1741.3

I hope this helps.