What is Hypothesis Testing? AP Statistics
Transcription
What is Hypothesis Testing? AP Statistics
AP Statistics Statistical Inference Hypothesis Testing What is Hypothesis Testing? A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses. The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected. Hypothesis Tests Statisticians follow a formal process to determine whether to reject a null hypothesis, based on sample data. This process, called hypothesis testing, consists of four steps…See Inference Toolbox (p. 705). Key Vocabulary (Text Chapters 11 – 13) Null Hypothesis (691) Confidence Intervals and Two-Sided Tests (710) Alternate Hypothesis (691) Type I and Type II Error (723) One-Sided Alternative (692) Power (729) Two-Sided Alternative (692) Power and Type II Error (729) Test Statistic (695) One-Sample t Statistic and t Distribution (744) p-Value (696) One-Sample t Test (746) Significance Level (699) One-Proportion z Test (766) Statistical Significance (699) Two-Sample Problems (781) Reject Ho (700) Conditions for Comparing Two Means (782) Fail to reject Ho (700) Two-Sample z Statistic (787) z-Test for a Population Mean (705) Two-Sample t Procedures (788) One-Sample z-Statistic Confidence Interval for Two Proportions (811) Significance Test for Two Proportions (815) (705) You should be familiar with all vocabulary related to hypothesis testing. AP Statistics Statistical Inference Hypothesis Testing Examples of Hypothesis Testing z-Test for a Population Mean (µ) At the bakery where you work, loaves of bread are supposed to weigh 1 pound. From experience, the weights of loaves produced at the bakery follow a Normal distribution with standard deviation = 0.13 pounds. You believe that new personnel are producing loaves that are heavier than 1 pound. As supervisor of Quality Control, you want to test your claim at the 5% significance level. You weigh 20 loaves and obtain a mean weight of 1.05 pounds. Hypotheses: Calculate the Test Statistic: ⁄√ p-Value = 1 – 0.9573 = 0.0427 (since we are testing µ > 1) Since σ is known, we will use z. α = 0.05 (5% significance level) (p < α) Solution/Interpretation: Since p < α, there is sufficient evidence against Ho. Therefore, reject Ho in favor of the alternate Ha. t – Test for a Population Mean (µ) An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. Suppose a simple random sample of 50 engines is tested. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of engines are normally distributed.) Hypotheses: Calculate the Test Statistic: ⁄√ Since σ is unknown, we will use t with df = 50 – 1 = 49 (use 40 df from Table C). For a One-Sample t, use 2P(T ≥ |t|) to calculate the value of p. The p-value will be between 2(0.025) and 2(.05) which gives us 0.05 < p < 0.1. For α = 0.05 (5% significance level), p is not less than α. Solution/Interpretation: Since p > α, there is not sufficient evidence against Ho. Therefore, we cannot reject Ho in favor of the alternate Ha. AP Statistics Statistical Inference z-Test for a Population Mean (µ) Hypothesis Testing (2-Sided Test and a Confidence Interval) Your friend uses Minitab to generate 25 observations at random from a Normal distribution with known mean and standard deviation. Unfortunately, he forgot to save the file. He remembers that the standard deviation was 4, and he thinks that the mean was 20. Before he closed the program, your friend did manage to print the following output. Descriptive Statistics: rand Variable N Mean Median TrMean StDev SE Mean 25 18.792 18.663 18.791 3.241 0.648 Minimum Maximum Q1 Q3 11.192 26.422 17.223 21.128 rand Variable rand Use an appropriate test to determine whether you believe your friend’s claim that the mean was 20. Hypotheses: Calculate a 95% Confidence Interval for µ: ( √ Since σ is known, we will use z. ) α = 0.05 (5% significance level) Solution/Interpretation: Since the hypothesized value falls within a 95% confidence interval, there is not sufficient evidence against Ho. Therefore, we cannot reject Ho in favor of the alternate Ha. Alternate Solution Hypotheses: Calculate the Test Statistic: ⁄√ p-Value = 2(0.0655) = 0.131 (since we are testing 2-sided) Since σ is known, we will use z. α = 0.05 (5% significance level) (p < α) Solution/Interpretation: Since p > α, there is not sufficient evidence against Ho. Therefore, we cannot reject Ho in favor of the alternate Ha. AP Statistics Statistical Inference Hypothesis Testing The amount of lead in a certain type of soil, when released by a standard extraction method, averages 86 parts per million (ppm). A new extraction method is tried on 40 specimens of the soil, yielding a mean of 83 ppm lead and a standard deviation of 10 ppm. Is there significant evidence at the 5% level that the new method frees less lead from the soil? Carry out an appropriate test to help answer this question. Hypotheses: Solution/Interpretation: Since p < α, there is sufficient evidence against Ho. Therefore, reject Ho in favor of the alternate Ha.
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