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Hypothesis Testing for Means and Proportions: Examples and Guidelines

This section discusses hypothesis testing for means (small samples) using t-distribution and proportions using z-test. It covers guidelines for conducting the tests, finding critical values, making decisions, and interpreting results. The examples focus on testing claims about means and proportions in various contexts.

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Hypothesis Testing for Means and Proportions: Examples and Guidelines

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  1. Section 7.3 Hypothesis Testing for the Mean (Small Samples)

  2. Similar to section 7.2… When the distribution is normal (or nearly normal), n < 30 or σ is unknown. • Use the t-distribution table: • Degrees of freedom: d.f. = n – 1

  3. EX: find critical value, t0 • Left tailed test, α = 0.01, n = 13 • Right tailed test, α = 0.10, n =10 • Two tailed test, α = 0.05, n = 22

  4. Guidelines for the t-Test • find H0 and Ha • Identify the level of significance, α • Identify the degrees of freedom, d.f. • Find the critical value(s) using the table. • Sketch the curve and shade the rejection region(s) • Find t • Make the decision to reject or not reject H0 • Interpret the decision in context.

  5. Use a t-test to test the claim • 14. Claim: µ > 25, α = 0.05, sample mean = 26.2, s = 2.32, n = 17 • 15. Claim: µ > 8000, α = 0.01, sample mean = 7700, s = 450, n = 25

  6. 18. A company claims that the mean battery life of their MP3 player is at least 30 hours. You suspect that the claim is incorrect and find that a random sample of 18 MP3 players has a mean battery life of 28.5 hours and a standard deviation of 1.7 hours. Is there enough evidence to reject the claim at alpha = 0.01?

  7. A repair shop believes that people travel more than 3500 miles between oil changes. A random sample of 8 cars getting an oil change has a mean distance of 3375 miles since the last oil change with a standard deviation of 225 miles. At alpha = 0.05, do you have enough evidence to support the shop’s claim?

  8. Section 7.4 HYPOTHESIS TESTING FOR PROPORTIONS

  9. Uses the z-Test

  10. Guidelines for the z-Test • np and nq both must be at least 5 • 1. find H0 and Ha • 2. identify α • 3. find the critical value(s) • 4. shade the rejection region(s) • 5. calculate z • 6. make decision to reject or not reject the null hypothesis • 7. interpret decision in context

  11. Determine if a normal distribution can be used. If so, test the claim.

  12. A research center claims that 16% of US adults say that curling is the Winter Olympic sport they would like to try the most. In a random sample of 300 US adults, 20% say that curling is the Winter Olympic sport they would like to try the most. At α = 0.05, is there enough evidence to reject the researcher’s claim?

  13. A research center claims that at most 75% of US adults think that drivers are safer using hands-free cell phones instead of using hand-held cell phones. In a random sample of 150 US adults, 77% think that drivers are safer using hands-free cell phones instead of hand-held cell phones. At α = 0.01, is there enough evidence to reject the center’s claim?

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