뒤로Hypothesis Testing for a Population Proportion – Step-by-Step Study Guide
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Q1. Hankook Tire Gauge Index Survey: Testing a Claim About a Proportion
Background
Topic: Hypothesis Testing for a Population Proportion
This question asks you to analyze survey data and test a claim about the proportion of adults who rate themselves as above average drivers. You will identify sample statistics, set up hypotheses, and understand the use of the normal approximation method for proportions.
Key Terms and Formulas
Sample proportion ():
Population proportion (): The value claimed or hypothesized for the population.
Test statistic for a proportion:
Step-by-Step Guidance
Find the actual number of respondents who rated themselves as above average drivers by multiplying the sample size by the given percentage.
Calculate the sample proportion () and identify the symbol.
Identify the value used for the population proportion () and the symbol that represents it.
Set up the null and alternative hypotheses for the test (but do not solve yet).
Try solving on your own before revealing the answer!
Final Answer:
A) (so, 877 respondents rated themselves as above average drivers)
B) Sample proportion:
C) Population proportion:
We use for the sample proportion and for the population proportion in hypothesis testing.
Q2. Lipitor Clinical Trial: Interpreting a 1-PropZTest Output
Background
Topic: Interpreting Hypothesis Test Output for Proportions
This question involves interpreting the output from a calculator for a hypothesis test about the proportion of subjects experiencing headaches in a clinical trial. You will identify the type of test, the test statistic, and the P-value.
Key Terms and Formulas
Test statistic (): Measures how many standard deviations the sample proportion is from the hypothesized proportion.
P-value: The probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.
Left-tailed, right-tailed, two-tailed: Refers to the direction of the alternative hypothesis.
Step-by-Step Guidance
Look at the alternative hypothesis in the calculator output (prop < 0.1) to determine the test direction.
Identify the test statistic () from the output.
Find the P-value from the output.
Write the null hypothesis and consider what the P-value means for the null hypothesis (but do not state the final conclusion yet).
Try solving on your own before revealing the answer!
Final Answer:
A) The test is left-tailed.
B) Test statistic:
C) P-value:
D) Null hypothesis: . Since the P-value is very small, we reject the null hypothesis.
E) There is sufficient evidence to support the claim that less than 10% of treated subjects experience headaches.
Q3. USA Today Survey: Testing a Claim About Biometric Security
Background
Topic: Hypothesis Testing for a Proportion (Calculator Output Interpretation)
This question asks you to interpret calculator output for a hypothesis test about the proportion of people who support replacing passwords with biometric security.
Key Terms and Formulas
Sample proportion (): The proportion observed in the sample.
Test statistic (): Standardized value for the sample proportion.
P-value: Probability of observing the sample result under the null hypothesis.
Critical value: The cutoff value for rejecting the null hypothesis at a given significance level.
Step-by-Step Guidance
Determine the direction of the test based on the claim (is it two-tailed, left-tailed, or right-tailed?).
Identify the test statistic () from the output.
Find the P-value from the output.
Write the null hypothesis and consider what the P-value means for the null hypothesis (but do not state the final conclusion yet).
Try solving on your own before revealing the answer!
Final Answer:
A) The test is two-tailed.
B) Test statistic:
C) P-value:
D) Null hypothesis: . Since the P-value is greater than 0.05, we do not reject the null hypothesis.
E) There is not sufficient evidence to warrant rejection of the claim that half of us say that we should replace passwords with biometric security.
Q4. MythBusters Toast Experiment: Testing Fairness of Outcomes
Background
Topic: Hypothesis Testing for a Proportion (Binomial Outcomes)
This question involves testing whether buttered toast lands buttered side down 50% of the time, using a significance level of 0.05.
Key Terms and Formulas
Null hypothesis ():
Alternative hypothesis ():
Test statistic:
P-value: Probability of observing the sample result under .
Step-by-Step Guidance
State the claim and write the null and alternative hypotheses.
Identify the significance level ().
Calculate the sample proportion () and set up the formula for the test statistic.
Set up the calculation for the P-value based on the test statistic (but do not compute the final value yet).
Try solving on your own before revealing the answer!
Final Answer:
Step 1: Claim is that (toast lands buttered side down half the time).
Step 2: Significance level is .
Step 3: Test statistic is calculated using , where .
Step 4: P-value is found using the standard normal distribution.
Step 5: Do not reject .
Step 6: There is not sufficient evidence to warrant rejection of the claim that dropped toast will land with the buttered side down 50% of the time.
Q5. OxyContin Clinical Trial: Testing for a High Rate of Nausea
Background
Topic: Hypothesis Testing for a Population Proportion
This question asks you to test the claim that more than 20% of OxyContin users develop nausea, using clinical trial data and a 0.05 significance level.
Key Terms and Formulas
Null hypothesis ():
Alternative hypothesis ():
Test statistic:
P-value: Area to the right of the test statistic in the standard normal distribution.
Step-by-Step Guidance
State the claim and write the null and alternative hypotheses.
Identify the significance level ().
Calculate the sample proportion () and set up the formula for the test statistic.
Set up the calculation for the P-value based on the test statistic (but do not compute the final value yet).
Try solving on your own before revealing the answer!
Final Answer:
Step 1: Claim is that (more than 20% develop nausea).
Step 2: Significance level is .
Step 3: Test statistic is calculated using , where .
Step 4: P-value is found using the standard normal distribution.
Step 5: Do not reject .
Step 6: There is not sufficient evidence to support the claim that more than 20% of OxyContin users develop nausea. However, the observed proportion is high.
Q6. Pew Research Center Poll: Testing for Less Than Half Preference
Background
Topic: Hypothesis Testing for a Population Proportion
This question asks you to test the claim that fewer than half of Americans prefer to watch the news, using a 0.01 significance level.
Key Terms and Formulas
Null hypothesis ():
Alternative hypothesis ():
Test statistic:
P-value: Area to the left of the test statistic in the standard normal distribution.
Step-by-Step Guidance
State the claim and write the null and alternative hypotheses.
Identify the significance level ().
Calculate the sample proportion () and set up the formula for the test statistic.
Set up the calculation for the P-value based on the test statistic (but do not compute the final value yet).
Try solving on your own before revealing the answer!
Final Answer:
Step 1: Claim is that (fewer than half prefer to watch the news).
Step 2: Significance level is .
Step 3: Test statistic is calculated using .
Step 4: P-value is found using the standard normal distribution.
Step 5: Reject .
Step 6: There is sufficient evidence to support the claim that fewer than half of Americans prefer to watch the news rather than read or listen to it.
Q7. NFL Overtime Coin Toss: Testing for Fairness
Background
Topic: Hypothesis Testing for a Population Proportion (Fairness Test)
This question asks you to test the claim that the coin toss in NFL overtime games is fair, using a 0.05 significance level.
Key Terms and Formulas
Null hypothesis ():
Alternative hypothesis ():
Test statistic:
P-value: Probability of observing the sample result under .
Step-by-Step Guidance
State the claim and write the null and alternative hypotheses.
Identify the significance level ().
Calculate the sample proportion () and set up the formula for the test statistic.
Set up the calculation for the P-value based on the test statistic (but do not compute the final value yet).
Try solving on your own before revealing the answer!
Final Answer:
Step 1: Claim is that (coin toss is fair).
Step 2: Significance level is .
Step 3: Test statistic is calculated using .
Step 4: P-value is found using the standard normal distribution.
Step 5: Reject .
Step 6: There is sufficient evidence to warrant rejection of the claim that the coin toss is fair. The coin toss rule does not appear to be fair.
