AP Statistics Chapter 9: Testing a Claim
What two circumstances guide us in choosing a level of significance?
-How plausible is Ho? -What are the consequences of rejecting Ho?
Describe the three influences that must be verified before deciding on how many observations are needed in a study.
-Significance Level: how much protection do we want against a Type 1 error -Practical Importance: how large a difference between the hypothesized parameter value and the actual parameter value -Power:how confident do we want to be that our study will detect a difference of the size we think is important
What is the most common error in concluding p-values?
Failing to find evidence against Ho means only that the data are consistent with Ho, not that we have clear evidence that Ho is true.
On what evidence would we fail to reject the null hypothesis?
If the sample result is likely to occur, them we will fail to reject Ho.
What is a Type 1 error?
If we reject Ho when Ho is true
Which error is worse, Type 1 or Type 2?
It depends on the situation.
If a P-value is large, what do we conclude about the null hypothesis?
Large P-values fail to give convincing evidence against Ho because they say that the observed result is likely to occur by chance when Ho is true.
What are the three conditions for conducting a significance test for a population mean?
Random, Normal, and Independent. ***normal=n≥30, OR examine sample data for normalness
Describe the four step process for significance tests and explain what is required at each step.
STATE: What hypotheses do you want to test, and at what significance level? Define any parameters you use. PLAN: Choose the appropriate inference method. Check conditions. DO: If conditions are met, perform calculations. -Compute the test statistic. -Find the p-value. CONCLUDE: Interpret the results of your test in the context of the problem.
What happens when the data does not support Ha?
Then we fail to reject the Ho
How do you calculate p-values using the t-distributions?
Use table B (after finding the test statistic), df=n-1, and the symmetry of the t distributions
If asked to carry out a significance test and there is no α provided, what is reccomended?
Use the default significance level, α=.05
How small should the P-value be in order to claim that a result is statistically significant?
smaller than α
How is the "DO" step different for a paired data question?
Instead of a random sample, we check random order. Normal is approximately the same. Assume independence between individual subjects, no need for 10% condition because this isn't sampling.
When using a fixed significance level to draw a conclusion in a statistical test what can be concluded when the P value is <α and ≥α?
P-value<α→reject Ho→conclude Ha (in context) P-value≥α→fail to reject Ho→cannot conclude Ha (in context)
If a P-value is small, what do we conclude about the null hypothesis?
Small P-values are evidence Against Ho because they say that the observed result is Unlikely to occur when Ho is true.
Two-sided alternative hypothesis
States that the parameter is different from the null hypothesis value (could be either larger or smaller). Sign used:≠
One-sided alternative hypothesis
States that the parameter is larger than the null hypothesis value or states that the parameter is smaller than the null value. Signs used:<>
What is paired data?
Study designs that involve making two observations on the same individual, or one observation on each of two similar individuals.
State the general form of the test statistic
Test statistic = (statistic-parameter)/(standard deviation of statistic)
Alternative Hypothesis
The claim about the population that we are trying to find evidence For. Notation: Ha
Null Hypothesis
The claim tested by a statistical test. The test is designed to assess the strength of the evidence Against the null hypothesis. Often the null hypothesis is a statement of "no difference." Notation: H₀ (Ho)
In any significance test...
The null hypothesis has the form Ho: parameter=value The alternative hypothesis has one of the forms Ha: parameter<value Ha: parameter>value Ha: parameter≠value
What is meant by the power of a significance level?
The power of a test against a specific alternative is the probability that the test will reject Ho at a chosen significance level α when the specified alternative value of the parameter is true.
What is the relationship between Power and Type 2 error? Will you be expected to calculate the power on the AP exam?
The power of a test against any alternative is 1 minus the probability of a Type 2 error for that alternative; that is. power=1-β
P-value
The probability, computed assuming Ho is true, that the statistic (such as p hat or x bar) would take a value as extreme or more extreme than the one actually observed. The smaller the P-value, the stronger the evidence Against Ho provided by the data.
What is the relationship between the significance level α and the probability of Type 1 error?
The significance level α of any fixed level test is the probability of a Type 1 error. That is, α is the probability that the test will reject the null hypothesis Ho when Ho is in fact true. Consider the consequences of a Type 1 error before choosing a significance level.
When using technology for the "DO" part of the four step process, what is recommended on page 573?
The textbook recommends doing the calculation with the appropriate formula then checking with the calculator. If you opt for the calculator-only method, name the procedure, report the test statistic, degrees of freedom, and the P-value.
What test statistic is used when testing for a population proportion? Is this on the formula sheet?
*formula not on formula sheet
For a one-sample t-test for a population mean, state:
-H₀: µ=µ₀ -Ha: µ>µ₀, µ<µ₀, µ≠µ₀ -
Summarize the three conditions that must be checked before carrying out significance tests.
-Normal: FOR P_ np and n(1-p)≥10 FOR µ_ n≥30, or stated normal -Random: usually stated -Independent (10%): p<1/10 Population (or stated independent)
What is meant by significance level?
A fixed value written as α (alpha) that determines whether or not we can reject the null hypothesis.
Significance Test
A formal procedure for comparing observed data with a claim (hypothesis) whose truth we want to assess. *about a population parameter
What does the test statistic measure? Is it on the AP exam formula sheet?
A test statistic measures how far a sample statistic diverges from what we would expect if the null hypothesis Ho were true, in standardized units. AND YES it is on the AP exam formula sheet.
Summarize the one-sample z test for a proportion.
Choose an SRS of size n from a large population that contains an unknown proportion p of success. To test the hypothesis Ho: p=p₀, compute the z statistic (in picture) Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction specified by the alternate hypothesis Ha. Use this test only when the expected numbers of success and failures np₀ and n(1-p₀) are both at least 10 and the population is at least 10 times as large as the sample. If Normality is not met, or if the population is less than 10 times as large as the sample, other procedures should be tested.
Describe the four points to be aware of when interpreting significance tests.
-Statistical Significance and Practical Importance: outliers can destroy the significance of convincing data -Don't Ignore Lack of Significance: large sample sizes are needed to come to important conclusions -Statistical Inference Is Not Valid for All Sets of Data: badly designed surveys/experiments often produce invalid results; random sampling/assignment ensure laws of probability. -Beware of Multiple Analyses: once you have a hypothesis, design a study to search specifically for the effect you now think is there.
How do you find p-values when carrying out a significance test about a population mean on the calculator?
2nd vars→tcdf. Insert lower, upper, df. Write as: tcdf(lower, upper, df)
In terms of rejecting the hypothesis H₀, how is a significance test related to a confidence interval on the same population?
???
Why don't we always use confidence intervals?
Confidence intervals only work for two-sided tests. There is only a useful connection between confidence intervals and two-sided tests.
Lets keep things straight.
Hypotheses always refer to a population, not a sample. Be sure to state Ho and Ha in terms of population parameters.
On what evidence would we reject the null hypothesis?
If our sample result is too unlikely to have happened by chance assuming Ho is true, then we'll reject Ho.
Explain what it means to say that data are statistically significant.
If the P-value is smaller than alpha, we say that the data are statistically significant at level α. In that case, we reject the null hypothesis Ho and conclude that there is convincing evidence in favor of the alternative hypothesis Ha. *not likely to happen by chance
What information would lead us to apply a paired t-test o a study, and what would be the statistic of interest?
If the conditions for inference are met, we can use paired t procedures to perform inference about the mean difference µd
What is a Type 2 error?
If we fail to reject Ho when Ho is false
When do you multiply the tcdf by 2?
When the Ha is two-sided
Can you use confidence intervals to decide between two hypotheses? What is the advantage to using confidence intervals for this purpose?
Yes, because the interval gives us the values of p that are consistent with the sample data.
What do you do if the degrees of freedom you need is not in Table B?
You use the next lower df that is available. *rounding up gives a Type 1 error