A random sample of 189 full-grown lobsters had a mean weight of 17 ounces and a standard deviation of 3.9 ounces. Construct a 95% confidence interval for the population mean.

Answer 1

Degrees of freedom = n-1 = 188

Critical value = 1.973

Confidence Interval: 17 +/- 1.973*3.9/sqrt(189)

=(16.44,17.56)

Please feel free to call/email me with questions.

Answer 2

Hello Bella S.,

When you submit a statistics question online, we need to know how you do computations in your class. Do you use a table and formulas or a statistical calculator, or computer software?

In my classes, we use a TI-83 or 84 calculator. Then press the STAT button, select Tests. The confidence intervals go from #7 to letter B. Your problem has one sample and the standard deviation comes from the data (sample standard deviation s) so you would use a t statistic. This is choice 8 on the calculator. After you choose that, make sure the word Stats is highlighted. Then you simply type in the 4 numbers in the problem (95% would be 0.95) and choose calculate. See if you can do this or if you don't use a calculator, resubmit the question with more details about which method your class uses for computations. Also send in any work that you have done or if you have the answer, send that in and we can check it.

Answer 3

For a confidence interval for a mean, we need four things. Luckily, the problem explicitly gives us 3 of the 4 things. Those four things are:

1. Sample mean (xbar)
2. Sample Standard Deviation (s)
3. Sample Size (n)
4. Normal quantile associated with our desired confidence level (z)

Here's what we've been given:

1. xbar = 17
2. s = 3.9
3. n = 189
4. confidence level = 95%, i.e. alpha = 0.05. These are equivalent statements. To get z, we can look it up in a standard normal quantile table, such as this one http://cs.ru.nl/~tomh/onderwijs/dm/dm_files/normtable.pdf.

In tables like this, the numbers in the middle are probabilities (area under the curve), and the numbers on the left and top are associated with standard normal quantiles (a.k.a "z-scores"). When constructing a confidence interval, we want the area under the curve to be equal to our confidence level, which means we want half of alpha on each tail end. So to get our z-score, we look up the z-score associated with an area of alpha/2 in the right tail.

alpha/2 = 0.025. Since there is 0.025 to the right of our z-score, that means there is 0.975 to the left of our z-score. So we look in the table and find where it says "0.975" or as close to that as possible, somewhere in the numbers in the middle. We find it, and all the way to the left it says "1.9", and all the way at the top it says "0.06," so our z-score is 1.96.

Now we have everything we need to construct our confidence interval. It will simply be:

xbar ± z * s/√n

we plug in everything we know and we get:

17 ± 1.96 * 3.9/√189

17 - 1.96 * 3.9/√189 ≈ 16.444

17 + 1.96 * 3.9/√189 ≈ 17.556

So our 95% confidence interval is (16.444, 17.556)