STA 2023 Exam 3
In a confidence interval for the difference between two population means has a positive lower-bound number and a positive upper-bound number, what can we conclude?
(+,+): 0 is not included in the confidence interval, so this is not plausible that the means are the same. Therefore, there is statistically significant evidence of a difference between two means. Moreover, you can conclude that μ1 is greater than μ2. (If μ1>μ2, then μ1-μ2>0.) If the confidence interval is (X,Y) then you can conclude that μ1 is between X greater and Y greater than μ2.
If a confidence interval for the difference between two population proportions has a positive lower-bound number and a positive upper-bound number, what can we conclude?
(+,+): 0 is not included in the confidence interval, so this is not plausible that the proportions are the same. Therefore, there is statistically significant evidence of a difference between two proportions. Moreover, you can conclude that p1 is greater than p2. (If p1>p2, then p1-p2>0.) If the confidence interval is (X,Y) then you can conclude that p1 is between X greater and Y greater than p2.
If a confidence interval for the difference between two population proportions has a negative lower-bound number and a positive upper-bound number, what can we conclude?
(-,+): 0 is in the confidence interval. Therefore, it is plausible that the proportions are the same. (If p1=p2, then p1-p2=0.) Therefore, there is no statistically significant evidence of a difference between the two proportions.
If a confidence interval for the difference between two population means has a negative lower-bound number and a positive upper-bound number, what can we conclude?
(-,+): 0 is in the confidence interval. Therefore, it is plausible that the proportions are the same. (If μ1=μ2, then μ1-μ2=0.) Therefore, there is no statistically significant evidence of a difference between the two means.
In a confidence interval for the difference between two population means has a negative lower-bound and a negative lower-bound number, what can we conclude?
(-,-): 0 is not included in the confidence interval, so this is not plausible that the means are the same. Therefore, there is statistically significant evidence of a difference between two means. Moreover, you can conclude that μ1 is less than μ2. (If μ1<μ2, then μ1-μ2<0.) If the confidence interval is (-X,-Y) then you can conclude that μ1 is between Y less than and X less than μ2.
If a confidence interval for the difference between two population proportions has a negative lower-bound number and a negative upper-bound number, what can we conclude?
(-,-): 0 is not included in the confidence interval, so this is not plausible that the proportions are the same. Therefore, there is statistically significant evidence of a difference between two proportions. Moreover, you can conclude that p1 is less than p2. (If p1<p2, then p1-p2<0.) If the confidence interval is (-X,-Y) then you can conclude that p1 is between Y less than and X less than p2.
If a two-sided significance test has a p-value greater than α (and thus you fail to reject H0), a ______% confidence interval will contain the value of H0.
(1-α)
What is the general equation for the test statistic?
(est. - #H0)/stderror
What is the estimator for a confidence interval for the difference between two population proportions?
(p̂1-p̂2)
Confidence Interval for the Difference between Two Population Proportions
(p̂1-p̂2)±z(√(p̂1(1-p̂1)/n1+p̂2(1-p̂2)/n2)
What is the estimator in a confidence interval for comparing two independent means?
(x̅1-x̅2)
Confidence Interval for Comparing Two Independent Means
(x̅1-x̅2)±t(√(s1^2/n1+s2^2/n2))
What is the standard error of the estimator when comparing two independent means?
(√(s1^2/n1+s2^2/n2))
A sample size of 20 has ______ degrees of freedom.
19
In a significance test with a α=0.05 level of significance, the probability of a Type I error is . . .
5%
Suppose you are constructing a confidence interval to compare two independent means. You take two samples: one with n=8 and one with n=10. How many degrees of freedom should you use in your analysis?
7 We always use the smaller of (n1-1) and (n2-1) as our degrees of freedom. ["Conservative approach"]
If Group 1 had three successes out of four observations and Group 2 has five successes out of eight observation, the pooled proportion would be . . .
8/12=0.67
The results of a two-sided significance test wil α=0.05 will agree with a ______% confidence interval.
95
One-Sided Hypothesis Test
A hypothesis test in which the values for which we can reject the null hypothesis (H0) are located entirely in one tail of the probability distribution. That is, the alternative hypothesis has a less-than symbol (<) or a greater-than symbol (>).
Two-Sided Hypothesis Test
A hypothesis test in which the values for which we can reject the null hypothesis (H0) are located in both tails of the probability distribution. That is, the alternative hypothesis has an "is not equal to" symbol (≠).
What does a low p-value mean?
A low p-value means that it would be extremely unlikely for us to get the sample statistic we did if the null hypothesis were true. It is more likely that the null hypothesis is not true, so we should reject it in favor of the alternative hypothesis.
McNemar's Test
A test used to compare proportions with matched-pairs data.
In a significance test with a α=0.05 level of significance, the probability of a Type II error is . . .
Cannot be determined. α is the probability of a Type I error, not a Type II error. We do not discuss how to calculate the probability of a Type II error in this class.
T/F: The results of a one-sided significance test and a confidence interval must always agree.
False.
T/F: Type I error accounts for errors due to survey design, sampling variability, and undercoverage.
False. It ONLY accounts for sampling variability. That is, it only accounts for the variability that we get because our samples are a subset of the larger population.
T/F: Statistically significant results are, by definition, practically significant.
False. Statistical significance and practical significance are separate concepts. A statistically significant result may or may not be practically significant.
T/F: If the p-value is high in a hypothesis test, we accept the null hypothesis.
False. We never accept the null hypothesis. We can only reject or fail to reject it.
What are the null and alternative hypotheses for a significance test comparing two independent proportions?
H0: p1-p2=0 Ha: p1-p2(<,>,≠)0
What are the hypotheses in a significance test comparing two independent means?
H0: μ1-μ2=0 Ha: μ1-μ2>0 (μ1>μ2) Ha: μ1-μ2<0 (μ1<μ2) Ha: μ1-μ2≠0 (μ1≠μ2)
What are the null and alternative hypotheses in a significance test for means?
H0: μ=# Ha: μ(<,>,≠) #
What are the hypotheses in a significance test comparing two dependent means?
H0: μd=0 Ha: μd (>,<,≠)0
If a two-sided significance test has a p-value ______ than α (and thus you reject H0), a (1-α)% confidence interval will not contain the value of H0.
Less.
What kind of error occurs when we reject the null hypothesis when it is really false?
No error is committed because this is not an error. We WANT to reject the null when it is false.
p-value
Represents the probability that the observed sample statistic value or an even more extreme sample statistic would occur, assuming the null hypothesis is true.
How is the test statistic calculated when using McNemar's test to compare two dependent proportions?
TS = YN-NY/√(YN+NY)
x̅d
The sample mean of the observed differences between two treatments in a dependent sample.
Type II Error
The type of error that occurs when we fail to reject the null hypothesis when the null hypothesis is really false.
Type I Error
The type of error that occurs when we reject the null hypothesis when the null hypothesis is really true.
T/F: Outliers can affect the validity of confidence intervals and significance tests.
True.
T/F: The results of a two-sided significance test and a confidence interval must always agree.
True.
What kind of error occurs when we reject the null hypothesis when it is really true?
Type I error
When do we use the statistical techniques for dependent samples (as opposed to those designed for independent samples)?
We use the techniques for dependent samples when treatments are given to the same experimental units, or two very similar ones. (e.g., Test the effectiveness of blood pressure medication on the same person.) We use the techniques for independent samples when treatments are given to different experimental units. (e.g., The amount of time males and females spend watching TV)
In a significance test for means, if the sample size is small and we are not told that the population is normally distributed, we should . . .
check the data for outliers by creating a dotplot or boxplot.
Degrees of Freedom
df=n-1
If the p-value>α, we . . .
fail to reject the null hypothesis at α level of significance.
The probability of committing a Type I error (decreases/increases) when the probability of committing a Type II error decreases.
increases
The discrepancy between statistical significance and practical significance is a particular concern when the sample size is ______.
large
A (low/high) p-value means that there is evidence against the null hypothesis and for the alternative hypothesis.
low
If the p-value is greater than 0.1, we say that there is ______ evidence against the null hypothesis and for the alternative hypothesis.
no statistically significant
The results of a hypothesis test are significant when . . .
p-value≥α.
Pooled Proportion
p̂= x1+x2/n1+n2
If the p-value<α, we . . .
reject the null hypothesis at α level of significance.
If the p-value is between 0.05 and 0.1, we say that there is ______ evidence against the null hypothesis and for the alternative hypothesis.
some
If the p-value is between 0.01 and 0.05, we say that there is ______ evidence against the null hypothesis and for the alternative hypothesis.
strong
In a significance test for means, the test statistic is a ______ score.
t
What is the test statistic for a significance test comparing two independent means?
t= (x̅1-x̅2)-0/√(s1^2/n1+s2^2/n2))
What is the test statistic in a significance test comparing two dependent means?
t= x̅d/(sd/√n)
What is the equation for a test statistic in a significance test for means?
t=(x̅ - #H0)/(s/√n))
If the p-value is between 0 and 0.01, we say that there is ______ evidence against the null hypothesis and for the alternative hypothesis.
very strong
Confidence Interval for Comparing Two Dependent Means (Matched Pairs)
x̅d±t(sd/√n)
Confidence Interval for a single population mean
x̅±t(s/√(n))
What is the test statistic for a significance test comparing two independent proportions?
z=(p̂1-p̂2)-0/√(p̂(1-p̂)(1/n1+1/n2))
Suppose that you could include all of the observations in a population in your sample. In this case, the probability of a Type I error would be . . .
zero.
Suppose that you could include all of the observations in a population in your sample. In this case, α would be . . .
zero.
The probability of committing a Type I error is equal to . . .
α, the level of significance
In hypothesis testing, what level of α corresponds with a 90% confidence level?
α=0.10
What are the assumptions for a significance test comparing two independent proportions?
• Categorical Data • SRS • There are at least 15 succ./fail. in each group
What are the assumptions in a significance test comparing two dependent means?
• Quantitative • SRS • n≥30, or the population was normally distributed (i.e., no outliers)
What are the assumptions in a significance test comparing two independent means?
• Quantitative data • SRS • Both sample sizes are at least 30 (n1≥30 and n2≥30), or the population was normally distributed (i.e., no outliers).
What assumptions must be met in a significance test for means?
• Quantitative data • SRS • n≥30 OR original population is Normal (If n<30 and you are not told that the original population is normally distributed, you should check for outliers. If there are outliers, you should conclude that the original population was NOT normally distributed.)
What is the standard error for a confidence interval for the difference between two population proportions?
√((p̂1(1-p̂1)/n1+p̂2(1-p̂2)/n2)
What is the standard error for a significance test comparing two proportions? (Express it using the pooled proportion.)
√(p̂(1-p̂)(1/n1+1/n2))