Confidence Intervals and Introduction to Inference Rapid Review for AP Statistics

Review the following concepts if necessary:

• Estimation and Confidence Intervals for AP Statistics
• Confidence Intervals for Means and Proportions for AP Statistics
• Sample Size for AP Statistics
• Statistical Significance and P-Value for AP Statistics
• The Hypothesis-Testing Procedure for AP Statistics
• Type-I and Type-II Errors and the Power of a Test for AP Statistics

Rapid Review

1. True–False. A 95% confidence interval for a population proportion is given as (0.37, 0.52). This means that the probability is 0.95 that this interval contains the true proportion.

Answer: False. We are 95% confident that the true proportion is in this interval. The probability is 0.95 that the process used to generate this interval will capture the true proportion.

2. The hypothesis that the Giants would win the World Series in 2002 was held by many of their fans. What type of error has been made by a very serious fan who refuses to accept the fact that the Giants actually lost the series to the Angels?

Answer: The hypothesis is false but the fan has failed to reject it. That is a Type-II error.

3. What is the critical value of t for a 99% confidence interval based on a sample size of 26?

Answer: From the table of t distribution critical values, t* = 2.787 with 25 df. Using a TI-84 with the invT function, the answer is given by invT(0.995,25). The 99% interval leaves 0.5% = 0.005 in each tail so that the area to the left of t* is 0.99 + 0.005 = 0.995.

4. What is the critical value of z for a 98% confidence interval for a population whose standard deviation we know?

Answer: This time we have to use the table of standard normal probabilities, Table A. If C = 0.98, 0.98 of the area lies between z* and –z*. So, because of the symmetry of the distribution, 0.01 lies above z*, which is the same as saying that 0.99 lies to the left of z*. The nearest table entry to 0.99 is 0.9901, which corresponds to z* = 2.33. Using the invNorm function on the TI-83/84, the answer is given by invNorm(0.99).

5. A hypothesis test is conducted with α = 0.01. The P-value is determined to be 0.037. Because the P-value > α, are we justified in rejecting the null hypothesis?

Answer: No. We could only reject the null if the P-value were less than the significance level. It is small probabilities that provide evidence against the null

6. Mary comes running into your office and excitedly tells you that she got a statistically significant finding from the data on her most recent research project. What is she talking about?

Answer: Mary means that the finding she got had such a small probability of occurring by chance that she has concluded it probably wasn't just chance variation but a real difference from expected.

7. You want to create a 95% confidence interval for a population proportion with a margin of error of no more than 0.05. How large a sample do you need?

Answer: Because there is no indication in the problem that we know about what to expect for the population proportion, we will use P* = 0.5. Then,

.

You would need a minimum of 385 subjects for your sample.

8. Which of the following statements is correct?
   1. The t distribution has more area in its tails than the z distribution (normal).
   2. When constructing a confidence interval for a population mean, you would always use z* rather than t* if you have a sample size of at least 30 (n > 30).
   3. When constructing a two-sample t interval, the "conservative" method of choosing degrees of freedom (df = min {n₁ – 1 n₂ – 1}) will result in a wider confidence interval than other methods.

Answer:

• I is correct. A t distribution, because it must estimate the population standard deviation, has more variability than the normal distribution.
• II is not correct. This one is a bit tricky. It is definitely not correct that you would always use z* rather than t* in this situation. A more interesting question is could you use z* rather than t*? The answer to that question is a qualified "yes." The difference between z* and t* is small for large sample sizes (e.g., for a 95% confidence interval based on a sample size of 50, z* = 1.96 and t* = 2.01) and, while a t interval would have a somewhat larger margin of error, the intervals constructed would capture roughly the same range of values. In fact, many traditional statistics books teach this as the proper method. Now, having said that, the best advice is to always use t when dealing with a one-sample situation (confidence interval or hypothesis test) and use z only when you know, or have a very good estimate of, the population standard deviation.
• III is correct. The conservative method (df = min{n₁ – 1, n₂ – 1}) will give a larger value of t*, which, in turn, will create a larger margin of error, which will result in a wider confidence interval than other methods for a given confidence level.