chapter 9: significance tests
how to find the test statistic
(statistic-parameter)/(standard deviation of statistic)
null hypothesis
H0: parameter=value The claim tested by a significance test
alternative hypothesis
Ha: value can be greater than/less than/not equal to the parameter The claim about the population we're trying to find evidence for
t-distribution
df=n-1; use when standard deviation of population is unknown
Which of the following 95% confidence intervals would lead us to reject H0: p = 0.30 in favor of Ha: p ≠ 0.30 at the 5% significance level? a. (0.27, 0.31) b. (0.24, 0.30) c. None of these d. (0.29, 0.38) e. (0.19, 0.27)
e. (0.19, 0.27)
Type II Error
fail to reject the null when it's false (has a probability of 1-beta)
interpreting p-value
if the null hypothesis of (context) is indeed true, there is (p-value) chance we obtained a result as extreme as (sample value) due to random chance alone
Normal conditions for t-distributions
Need a sample size of at least 30 to assume Normality without work. If sample size is too small, data needs to be graphed to determine skewness/outliers
p-value
the probability that measures the strength of the evidence against a null hypothesis
the smaller the p-value...
the stronger the evidence against the null provided by the data
a _____ ______ test at significance level 0.05 and a 95% confidence interval gives similar information about the population parameter
two-sided
the 95% confidence interval gives an approximate range of p0's that would not be rejected by a _____ ______ test at the 0.05 significance level
two-sided
an appropriate 95% confidence interval for µ has been calculated as (-0.73, 1.92) based on n=15 from a population with a Normal distribution. if we wish to use this confidence interval to test the hypothesis H0: µ = 0 against Ha: µ ≠ 0, what is a legitimate conclusion?
we cannot perform the required test since we do not know the value of the test statistic
when is the alternative hypothesis two-sided?
when it states the parameter is different from the null (not equal to)
when is the alternative hypothesis one-sided?
when it states the parameter is larger than the null hypothesis or if it states the parameter is smaller than the null
when do we reject the null?
when our sample result is too unlikely to have happened by chance--p-value is smaller than alpha (sufficient evidence to conclude the alternative)
when do we fail to reject the null?
when p-value is larger, or equal to, the alpha--there's insufficient evidence to conclude the alternative
when the two-sided test at a level (insert) fails to reject the null, the confidence level _______ contain the mean
will
when the two-sided significance test at level (insert) rejects the null, the confidence interval for the mean ______ _______ contain the hypothesized value of the mean
will not
A random sample of 100 likely voters in a small city produced 59 voters in favor of Candidate A. The observed value of the test statistic for testing the null hypothesis H0: p = 0.5 versus the alternative hypothesis Ha: p > 0.5 is...
z= (0.59-0.5)/sqrt ((0.5x0.5)/100)
After once again losing a football game to the archriveal, a college's alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was taken, and 64 of the alumni in the sample were in favor of firing the coach. Suppose you wish to see if a majority of living alumni are in favor of firing the coach. The appropriate test statistic would be...
z=(0.64-0.5)/sqrt((0.5x0.5)/100)
PHANTOMS
P: parameter H: hypothesis A: assess conditions N: name test T: test statistic O: obtain p-value M: make decision S: state conclusion
how does the value of the alternative parameter affect power?
the greater the difference between the hypothesized and true mean, the more obvious the result and, therefore, the greater the power
Are TV commercials louder than their surrounding programs? To find out, researchers collected data on 50 randomly selected commercials in a given week. With the televisions's volume at a fixed setting, they measured the maximum loudness of each commercial and the maximum loudness in the first 30 seconds of regular programming that followed. Assuming conditions for inference are met, the most appropriate method for answering the question of interest is...
a paired t test
Type I Error
reject the null when it's true (has a probability of alpha)
factors that affect power
sample size, alpha significance level, and value of the alternative parameter
A significance test allows you to reject a null hypothesis in favor of an alternative at the 5% significance level. What can you say about significance at the 1% level? a. The answer cannot be determined from the information given. b. H0 can be rejected at the 1% significance level. c. Ha can be rejected at the 1% significance level. d. There is insufficient evidence to accept H0 at the 1% significance level. e. There is sufficient evidence to accept Ha at the 1% significance level.
a. the answer cannot be determined from the information given
The most important condition for sound conclusions from statistical inference is that... a. the data come from a well-designed random sample or randomized experiment. b. the data contain no outliers. c. the sample size be no more that 10% of the population size. d. the sample size be at least 30. e. the population distribution be exactly Normal.
a. the data come from a well-designed random sample or randomized experiment
After checking that conditions are met, you perform a significance test of H0: μ = 1 versus µ ≠ 1. You obtain a P-value of 0.022. Which of the following is true? a. A 95% confidence interval for μ will include the value 0. b. A 99% confidence interval for μ will include the value 1. c. None of these is necessarily true. d. A 99% confidence interval for μ will include the value 0. e. A 95% confidence interval for μ will include the value 1.
b. a 99% confidence interval for μ will include the value 1
Which of the following is not a condition for performing a significance test about an unknown population proportion p? a. The data should come from a random sample or randomized experiment. b. The population distribution should be approximately Normal, unless the sample size is large. c. Both np and n(1 - p) should be at least 10. d. If you are sampling without replacement from a finite population, then you should sample no more than 10% of the population. e. Individual measurements should be independent of one another.
b. the population distribution should be approximately Normal, unless the sample size is large
You are thinking of conducting a one-sample t test about a population mean μ using a 0.05 significance level. You suspect that the distribution of the population is not Normal and may be moderately skewed. Which of the following statements is correct? a. You can carry out the test only if the population standard deviation is known. b. You can safely carry out the test if your sample size is large and there are no outliers. c. You should not carry out the test because the population does not have a Normal distribution. d. The t procedures are robust-you can use them any time you want. e. You can safely carry out the test if there are no outliers, regardless of the sample size.
b. you can safely carry out the test if your sample size is large and there are no outliers
how does sample size affect power?
the larger the sample size, the higher the power
A 95% confidence interval for a population mean μ is calculated to be (1.7, 3.5). Assume that the conditions for performing inference are met. What conclusion can we draw for a test of H0: μ = 2 versus Ha: μ ≠ 2 at the 0.05 level based on the confidence interval? a. None. We cannot draw a conclusion at the 0.05 level since this test is connected to the 97.5% confidence interval. b. We would reject H0 at level 0.05. c. We would fail to reject H0 at level 0.05. d. None. Confidence intervals and significance tests are unrelated procedures. e. None. We cannot carry out the test without the original data.
c. we would fail to reject H0 at level 0.05
Vigorous exercise helps people live several years longer (on average). Whether mild activities like slow walking extend life is not clear. Suppose that the added life expectancy from regular slow walking is just 2 months. A statistical test is more likely to find a significant increase in mean life expectancy if... a. the size of the sample doesn't have any effect on the significance of the test. b. it is based on a very small random sample and a 1% significance level is used. c. it is based on a very small random sample and a 5% significance level is used. d. it is based on a very large random sample and a 5% significance level is used. e. it is based on a very large random sample and a 1% significance level is used.
d. it is based on a very large random sample and a 5% significance level is used
in a test of H0: µ=100 against Ha: µ ≠ 100, a sample size of 10 produces a sample mean of 103 and a pvalue of 0.08. which of the following is true at the 5% significance level? a. there's significant evidence to conclude µ ≠ 100 b. there's sufficient evidence to conclude that µ = 100 c. there's insufficient evidence to conclude that µ = 100 d. there's insufficient evidence to conclude that µ ≠ 100 e. there's sufficient evidence to conclude that µ is greater than 103
d. there's insufficient evidence to conclude that µ ≠ 100
The z statistic for a test of H0: p = 0.4 versus Ha: p > 0.04 is z = 2.43. This test is... a. not significant at either 0.05 or 0.01 b. inconclusive because we don't know the vlaue of p-hat. c. significant at 0.05 but not at 0.01 d. significant at 0.01 but not at 0.05 e. significant at both 0.05 and 0.01
e. significant at both 0.05 and 0.01
A researcher claims to have found a drug that causes people to grow taller. The coach of the basketball team at Brandom University has expressed interest but demands evidence. Over 1000 Brandom students volunteer to participate in an experiment to test this new drug. Fifty of the volunteers are randomly selected, their heights are measured, and they are given the drug. Two weeks later, their heights are measured again. The power of the test to detect an average increase in height of one inch could be increased by... a. using 0.01 instead of 0.05 b. giving the drug to 25 randomly selected students instead of 50 c. using only volunteers form the basketball team in the experiment d. using a 2 sided test instead of a 1 sided test e. using 0.05 instead of 0.01
e. using 0.05 instead of 0.01
how does alpha significance level affect power?
increasing alpha increases the power because a less conservative alpha increase the chance of correctly rejecting the null
when you increase power, what happens to the probability of a Type II error occurring?
it decreases
Normal conditions for proportions
np and n(1-p) must be greater than/equal to 10
the 90% confidence interval gives an approximate range of p's that would not be rejected by a ______ ______ test at the 0.05 significance level
one-sided
hypotheses always refer to the __________
population
