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Hypothesis Tests with Proportions

Explore hypothesis tests with proportions in Chapter 10, understanding distributions, significance, and calculations for statistical analysis. Learn to write hypotheses, conduct tests, and interpret results effectively.

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Hypothesis Tests with Proportions

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  1. Hypothesis Tests with Proportions Chapter 10

  2. Write down the first number that you think of for the following . . . Pick a two-digit number between 10 and 50, where both digits are ODD and the digits do not repeat.

  3. What possiblevalues fit this description? • Record your answer on the dotplot on the board. • What do you notice about this distribution? • Did you expect this to happen?

  4. What proportion of the time would I expect to get the value 37 if the values were equally likely to occur? • Is the difference in these proportions significant? A hypothesis test will help me decide! How do I know if this p-hat is significantly different from the 1/8 that I expect to happen?

  5. What are hypothesis tests? Is it one of the sample proportions that are likely to occur? These calculations (called the test statistic) will tell us how many standard deviations a sample proportion is from the population proportion! Calculations that tell us if the sample statistics (p-hat) occurs by random chance or not OR . . . if it is statistically significant Is it . . . • a random occurrence due to natural variation? • an occurrence due to some other reason? Statistically significant means that it is NOT a random chance occurrence! Is it one that isn’t likely to occur?

  6. Nature of hypothesis tests - How does a murder trial work? • First begin by supposing the “effect” is NOT present • Next, see if data provides evidence against the supposition Example: murder trial First - assume that the person is innocent Then – must have sufficient evidence to prove guilty Hmmmmm … Hypothesis tests use the same process!

  7. Notice the steps are the same as a confidence interval except we add hypothesis statements – which you will learn today Steps: • Assumptions • Hypothesis statements & define parameters • Calculations • Conclusion, in context

  8. Assumptions for z-test: Have an SRSof context Large Sample Size?...becausebothnp > 10andn(1-p) > 10 Population is at least 10n YEA – These are the same assumptions as confidence intervals!!

  9. Check assumptions for the following: • Given SRS of homes • Large Sample size? np=150 & n(1-p)=350 (both are greater than 10) • There are at least 5000 homes in the county. Example 1: A countywide water conservation campaign was conducted in a particular county. A month later, a random sample of 500 homes was selected and water usage was recorded for each home. The county supervisors wanted to know whether their data supported the claim that fewer than 30% of the households in the county reduced water consumption after the conservation campaign.

  10. How to write hypothesis statements • Null hypothesis – is the statement (claim) being tested; this is a statement of “no effect” or “no difference” • Alternative hypothesis – is the statement that we suspect is true H0: Ha:

  11. How to write hypotheses: Null hypothesis H0: parameter = hypothesized value Alternative hypothesis Ha: parameter > hypothesized value Ha: parameter < hypothesized value Ha: parameter = hypothesized value

  12. Example 2:(Back to the opening activity) Is the proportion of students who answered 37 higher than the expected proportion of 1/8? H0: p = 1/8 Ha: p > 1/8 Where p is the true proportion of people who answered “37”

  13. Example 3: A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. Is this claim too high? H0: p = .7 Ha: p < .7 Where p is the true proportion of vaccinated people who do not get the flu

  14. Example 4: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses : H0: m = 40 Ha: m = 40 Where m is the true mean amperage of the fuses

  15. Facts to remember about hypotheses: • Hypotheses ALWAYS refer to populations(use parameters – never statistics) • The alternative hypothesis should be what you are trying to prove! • ALWAYS define your parameter in context!

  16. Must use parameter (population) x is a statistics (sample) Activity: For each pair of hypotheses, indicate which are not legitimate & explain why Must be NOT equal! p is the population proportion! Must use same number as H0! P-hat is a statistic – Not a parameter!

  17. Calculations Once you have made your statements, you need to make calculations that will determine if you will ”fail to reject” or “reject” the null hypothesis.

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