# Statistics Questions

See attached file for full problem description.

Confidence Intervals

1. For a sample mean of 4 and a population standard deviation of 0.3 with a sample size of 100, find the 95% CI for the population mean.

2. Using Internet Explorer, go to http://www.rossmanchance.com/applets/Confsim/Confsim.html . (It may take a minute or two to load.) Fix your settings on the left-hand side so that they read: Method: Means; Z with sigma;  = 4; =0.3; n = 100; and Intervals = 1. Assume your true population mean () is 5. Hit the "Sample" button to get a single confidence interval. The solid vertical  line indicates your true mean.

Did your confidence interval from your single sample include the true population mean? (The applet makes the confidence interval green if it does hit the true population mean and red if it doesn't.)

What is this applet doing? It is taking samples of size 100 from a population with a mean of 5 and a standard deviation of 0.3. Not every sample of size 100 will have a sample mean close to the population mean.

3. If you have a 95% confidence interval, then if you take many samples of the same size, _______________________________________________________________.

(Hint: think about the definition of confidence interval)

4. What is the "true population mean"? (Define it, don't use any numbers.)

Remember that in real life, we almost never know the true population mean or population proportion or population standard deviation. We use inference (confidence intervals and hypothesis testing) to give us good estimates of the true population parameters.

5. Try changing intervals = 100 to show 100 separate confidence intervals. Do it again for 1000 separate confidence intervals. What % of your confidence intervals contain the true population mean for n = 100 and n = 1000?

6. Compare your confidence intervals for n = 10, n=100, and n = 1000.

n = 10 n = 100 (from #1) n = 1000

4  4  4  0.02

( , ) ( , ) (3.98, 4.02)

How do your confidence intervals change as n increases?

7. Return to a sample size of 100. Compare your confidence intervals for confidence levels of 90%, 95%, and 99%.

90% 95% (from #1) 99%

4  4  4  0.08

( , ) ( , ) (3.92, 4.08)

How do your confidence intervals change as the confidence level increases?

Hypothesis testing

For each of the stories below, state the hypotheses and neatly label the normal curve with the number and symbol for the mean you use in your null hypothesis (0), the sample mean ( ), the standard deviation (), z = 0, and the test statistic (z0). Also shade the appropriate part of the curve which shows the P-value. You do not need to calculate the P-value or conclude your test.

8. A study of the nutrition of dialysis patients measured the level of phosphorus in the blood of a patients on 6 occasions. The mean phosphorus level for this patient was 5.33 mg/dl and a population standard deviation for all dialysis patients of 0.9 mg/dl. Is there evidence that this patient's mean phosphorus level exceeds 4.8 mg/dl, the upper end of the "normal" range?

9. Reading test scores for a sample of 44 third-graders from Cumberland Elementary School in West Lafayette show a mean of 35.09. The meant test scores for all Indiana third-graders is 32, with a population standard deviation of 11. Do third-graders from Cumberland have a mean score different from that of the state of Indiana?

See attached file for full problem description.

#### Solution Preview

Confidence Intervals

1. For a sample mean of 4 and a population standard deviation of 0.3 with a sample size of 100, find the 95% CI for the population mean.

The formula for a confidence interval is:

+ z*

We use the z-statistic because we know the population standard deviation.

For this problem, x-bar is 4, sigma is 0.3, n is 100, and z* is 1.96 (the critical value for p = 0.05). So, plugging those in...

4 ± 1.96(0.3/√100)

4 ± 1.96(0.03)

4 ± 0.0588

(3.9412, 4.0588)

2. Using Internet Explorer, go to http://www.rossmanchance.com/applets/Confsim/Confsim.html . (It may take a minute or two to load.) Fix your settings on the left-hand side so that they read: Method: Means; Z with sigma;  = 4; =0.3; n = 100; and Intervals = 1. Assume your true population mean () is 5. Hit the "Sample" button to get a single confidence interval. The solid vertical  line indicates your true mean.

Did your confidence interval from your single sample include the true population mean? (The applet makes the confidence interval green if it does hit the true population mean and red if it doesn't.)

Mine did. This applet is drawing samples from the population mentioned in the previous problem. See the answer to the next question to see why it's more likely than not that the true population mean falls within the ...