Statistics chapter 9
An advertiser wishes to see if a new advertisement is effective in promoting an existing product. The previous advertisement has a recognition score of 3.7. An SRS of 12 potential buyers resulted in a mean recognition score of 3.4 with a standard deviation of 1.7. Which of the following required conditions for conducting a t-test for a mean has not been met?
The population is Normally distributed or n is large.
Which of the following statements is FALSE?
The power of a hypothesis test does not depend on the sample size.
If we reject the null hypothesis when, in fact, it is true, we have
committed a Type I error
Use Scenario 9-1. Which of the following assumptions for inference about a proportion using a hypothesis test are violated?
np>10
Use Scenario 9-2. The hypotheses for testing the teacher's claim are
p=0.2, p<0.2
The most important condition for sound conclusions from statistical inference is usually
that the data can be thought of as a random sample from the population of interest.
In testing hypotheses, if the consequences of incorrectly rejecting the null hypothesis are very serious, we should
use a very small level of significance.
You construct a 95% confidence interval for a mean and find it to be 1.1 ± 0.8. Which of the following is true?
A test of the hypotheses H0: = 0, Ha: 0 would reject H0 at the 0.05 level.
To determine if having children within the first two years of marriage increases the divorce rate, where p = proportion of marriages that end in divorce, we should test the hypotheses
P=0.5, P>0.5
Use Scenario 9-1. Suppose that only two of the apples sampled are found to have major defects, and she proceeds with the test. The P-value of her test is
greater than .10
In a statistical test of significance, we say the data are statistically significant at level if
the P-value is at most
If a significance test gives P-value 0.005
we do have convincing evidence against the null hypothesis.
I draw an SRS of size 15 from a population that has a normal distribution with mean and standard deviation . The one-sample t statistic has how many degrees of freedom?
14
We wish to see if the dial indicating the oven temperature for a certain model oven is properly calibrated. Four ovens of this model are selected at random. The dial on each is set to 300ºF, and, after one hour, the actual temperature of each is measured. The temperatures measured are 305º, 310º, 300º, and 305º. Assuming that the actual temperatures for this model when the dial is set to 300º are Normally distributed with mean , we test whether or not the dial is properly calibrated by testing the hypotheses H0: = 300, Ha: 300. Which of the following is the value of the test statistic for a one sample t-test?
2.45
You test the hypothesis against the alternative and obtain a P-value of 0.022. Which of the following must be true?
A 99% confidence interval for will include the value 1.
An agricultural researcher plants 25 plots with a new variety of corn. The average yield for these plots is bushels per acre. Assume that the yield per acre for the new variety of corn follows a Normal distribution with unknown mean and that a 95% confidence interval for is found to be 150 ± 3.29. Which of the following is true?
A test of the hypotheses H0: = 155, Ha: 155 will be significant at the 0.05 level.
Which of the following will increase the power of a statistical test of significance.
Increase the sample size.
Use Scenario 9-3. Which of the following best describes the sampling distribution of proportions for this test?
Mean = 0.43; Standard deviation = 0.064; shape approximately Normal
In 1999, 2.2% of all cars in the United States were reported stolen. In a random sample of 400 Nissan Maxima cars that year, 12 were reported stolen. Is this evidence (at the = 0.05 level) that the theft rate for this model is higher than the national rate?
No, the P-value = 0.1685, so we fail to reject H0 and cannot conclude that the rate for Nissans is higher that 2.2%
Suppose we are testing the null hypothesis H0: = 50 and the alternative Ha: 50 for an approximately Normal population.. A random sample of nine observations are drawn from the population and we find the sample mean of these observations is J = 53 with a standard deviation of 6.48. Which of the following intervals contains the P-value for this test?
P-value > 0.2
In formulating hypotheses for a statistical test of significance, the null hypothesis is often
a statement of "no effect" or "no difference."
Use Scenario 9-4. Which of the following intervals contains the P-value for this test?
between 0.05 and 0.01
A researcher plans to conduct a test of hypotheses at the = 0.01 significance level. She designs her study to have a power of 0.90 at a particular alternative value of the parameter of interest. The probability that the researcher will commit a Type I error is
0.01.
Use Scenario 9-2. The P-value for this test is closest to:
0.0125
A 40-year-old high school teacher who takes great pride in his youthful appearance asks the 75 students in his classes to guess his age. He assumes that his 75 students are a simple random sample of all high school students with respect to their estimates of his age. The 95% confidence interval for the students' age guesses is years. Which of the following conclusions can be drawn from this result?
A test of the hypotheses H0: = 40, Ha: 40 will be significant at the 0.05 level.
The average growth of a certain variety of pine tree is 10.1 inches in three years. A biologist claims that a new variety will have a greater three-year growth. A random sample of 25 of the new variety has an average three-year growth of 10.8 inches and a standard deviation of 2.1 inches. The appropriate null and alternative hypotheses to test the biologist's claim are
H0: µ = 10.1 against Ha: µ > 10.1
I wish to test the hypothesis based on an SRS of size n from a Normal population. I calculate a 95% confidence interval for and find it to be 1.33 to 4.67. Which of the following is true?
I would reject H0 at level .05.
Nine swimmers are randomly selected from a large group. Each is asked to hold their breath for as long as possible and the times are recorded. Then they are given instructions in a new method for relaxing while holding their breath. Afterwards, they are again timed on how long they can hold their breath. We wish to perform a t-test on this paired data to see if the swimmers held their breath for longer after receiving breath-holding instructions. Which of the following is not a required condition to perform a t-test on these paired data?
The distribution of the swimmers breath-holding times before the training is approximately Normal.
One effect of the pesticide DDT upon birds is to inhibit the production of the enzyme carbonic anhydrase, which controls calcium metabolism. It is believed that this causes eggshells to be thinner and weaker than normal and makes the eggs more prone to breakage. An experiment was conducted where 16 sparrow hawks were fed a mixture of 3 ppm dieldrin and 15 ppm DDT (a combination often found in contaminated prey). The first egg laid by each bird was measured, and the mean shell thickness was found to be 0.19 mm. A "normal" eggshell has a mean thickness of 0.2 mm. The null and alternative hypotheses are
U=0.2, U<0.2
In tests of significance about an unknown parameter of some population, which of the following is considered strong evidence against the null hypothesis?
We observe a value of an estimate of the unknown parameter based on a simple random sample from the population that is very unlikely to occur if the null hypothesis is true.
You are thinking of using a t procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect that the distribution of the population is not normal and may be moderately skewed. Which of the following statements is correct?
You may use the t procedure provided your sample size is at least thirty
The P-value of a test of a null hypothesis is the probability that
assuming the null hypothesis is true, the test statistic will take a value at least as extreme as that actually observed.
A noted psychic was tested for ESP. The psychic was presented with 200 cards face down and asked to determine if the card was one of five symbols: a star, cross, circle, square, or three wavy lines. The psychic was correct in 50 cases. To determine if he has ESP, we want to know if his success rate is better than someone who just guesses. That is, we test the hypotheses H0: p = 0.20, Ha: p > 0.20, where p represent the proportion of cards for which the psychic correctly identifies the symbol in random trials. Assume the 200 trials described above can be treated as an SRS from the population of all guesses the psychic would make in his lifetime. The P-value of this test is
between .05 and .025.
Use Scenario 9-4. Suppose the mean and standard deviation we obtained were based on a sample of 25 postal workers, rather than 100. The P-value would be
larger.
A university administrator obtains a sample of the academic records of past and present scholarship athletes at the university. The administrator reports that no significant difference was found in the mean GPA (grade point average) for male and female scholarship athletes (P = 0.287). This means that
the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample, if there is no difference in mean GPAs, is 0.287.
In a test of statistical hypotheses, the P-value tells us
the probability of obtaining a sample with a test statistic that is farther from 0 than the one we obtained.
In hypothesis testing, is the probability of committing a Type II error. The power of the test, , is then
the probability of rejecting H0 when Ha is true
The power of a statistical test of hypotheses is
the probability that a significance test will reject the null hypothesis when a particular alternative value of the parameter is true.
A level two-sided significance test rejects the null hypothesis H0: = 0 when
the value 0 falls outside a level 1- confidence interval for U (not Uo)
. Use Scenario 9-3. The test statistic, P-value, and appropriate decision for this test are:
z = 2.40; P-value = 0.008; reject Ho