An industrial engineer evaluating logic wafers was interested in the amount of silver required in the photographic stages. for 15 batches, the following net usages (ounces per wafer) were determined: 0.473 0.315 0.521 0.253 0.588 0.343 0.439 0.392 0.217 0.111 0.394 0.251 0.331 0.101 0.448 construct a 90% confidence interval for the mean silver usage per wafer. lower confidence bound= upper confidence bound=

The 90% confidence interval is given by (0.281;0.409)  

Lower Confidence bound =0.281

Upper Confidence Bound=0.409

Step-by-step explanation:

1) Notation and definitions  

n=15 represent the sample size  

represent the sample mean  

represent the sample standard deviation  

m represent the margin of error  

Confidence =90% or 0.90

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

2) Calculate the mean and standard deviation for the sample

On this case we need to find the sample standard deviation with the following formula:

And in order to find the sample mean we just need to use this formula:

The sample mean obtained on this case is and the deviation s=0.141

3) Calculate the critical value tc  

In order to find the critical value is important to mention that we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 90% of confidence, our significance level would be given by and . The degrees of freedom are given by:  

We can find the critical values in excel using the following formulas:  

"=T.INV(0.05,14)" for  

"=T.INV(1-0.05,14)" for  

The critical value  

3) Calculate the margin of error (m)  

The margin of error for the sample mean is given by this formula:  

4) Calculate the confidence interval  

The interval for the mean is given by this formula:  

And calculating the limits we got:  

The 90% confidence interval is given by (0.281;0.409)  

question 1 is 8, question 2 is 9, question 3 is 24

step-by-step explanation:

now, the numbers they give you at the top are the key. from there, you just have to plug and chug! let's try 1, 2, and 3!

key: a=3, b=5, c=6

1. a+5

remember, a=3, so we plug that in to get 3+5=8

2. 15-c

c=6, so we plug that in to get 15-6=9

3. 4b

b=6, so we plug it in to get 4*6=24

time to try the rest on your own! if you put the answers below in a comment, i'll check them for you!