(d) The probability that the company will change its delivery procedures(e) The probability that the company will fail to reject H (^0)
solution: The power of the test is the probability of rejecting a false null hypothesis in favor of an
alternative—in this case, the hypothesis that the company is on time 90% of the time is false. If
we correctly reject this hypothesis, the company will change its delivery procedures. Hence,
(d) is the correct answer.
Rapid Review
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. Because this interval has been created, there is no repeatable random event. That’s
why we say “We are 95% confident.” It avoids improper use of the word probability. (The
probability is 0.95 that the process used to generate this interval will capture the true proportion.)
- 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. - 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.
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 ) .
- 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. - 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. - 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