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Examples of Inference

Examples of Inference. Example Problem. Example: Runners. You measure the weights of 24 male runners. You do not actually choose an SRS, but you are willing to assume that these runners are a random sample from the population of male runners in your town. Here are their weights in kilograms:

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Examples of Inference

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  1. Examples of Inference Example Problem

  2. Example: Runners • You measure the weights of 24 male runners. You do not actually choose an SRS, but you are willing to assume that these runners are a random sample from the population of male runners in your town. Here are their weights in kilograms: 67.8 61.9 63.0 53.1 62.3 59.7 55.4 58.9 60.9 69.2 63.7 68.3 64.7 65.6 56.0 57.8 66.0 62.9 53.6 65.0 55.8 60.4 69.3 61.7

  3. Example: Runners • Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. • Step 1: First, state what you want to know in terms of the Parameter and determine what the question is asking • We wish to estimate the true mean weight, μ, of male runners in our town

  4. Example: Runners • Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. • Step 2: Second, examine theAssumptions and check the conditions. • Independence: • Randomization condition: We assumed that the 24 runners represented a random sample of the town. • 10% condition: We safely assume that there are more than 240 runners in this town • If there are not more than 240 runners or independence is violated, then our results may not be valid.

  5. Example: Runners • Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. • Step 2: Second, examine theAssumptions and check the conditions. • Normality: • Since the sample is small, we cannot apply the CLT. • We do not know if the population is normal. • We need to examine the sample data • Since the sample looks fairly normal, we can assume that the sampling model will look approximately normal.

  6. Example: Runners • Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. • Step 3: Third, Name the inference, do the work, and state the Interval • We will construct a 95% 1-sample t-Interval for means: • Where did 2.069 come from? • It is t* for 95% Confidence and 23 degrees of freedom from TABLE B

  7. Example: Runners • Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. • Step 4: Fourth, last but not least, state your Conclusion in context of the problem • We are 95% confident that the true mean weight of male runners in the town is between 59.762kg and 63.822kg. However, some of our assumptions may not be valid. If the assumptions are violated, then our results may not be legitimate.

  8. Example: Runners Part 2 • Suppose that the average weight of a runner in the city has traditionally been 64kg. The runners that you gathered in your sample were on a diet to enhance their metabolism. The diet claims that it will make the runners lighter and develop a leaner muscle mass. Did the diet work? Use a 95% confidence level. • If our confidence is 95%, what is α? • Is this one or two tailed? 0.05 This is one tailed

  9. Example: Runners Part 3 • Step 1: Identify population Parameter, state the null and alternative Hypotheses, determine what you are trying to do (and determine what the question is asking). • H0: The mean weights of the runners will be 64kg • HA: The mean weights of the runners will be less than 64kg

  10. Example: Runners • Step 2: Verify theAssumptionsby checking the conditions • Independence: • Randomization condition: We assumed that the 24 runners represented a random sample of the town. • 10% condition: We assume that there are more than 240 runners in this town • If there are not more than 240 runners or independence is violated, then our results may not be valid.

  11. Example: Runners • Step 2: Verify theAssumptionsby checking the conditions: • Normality • Since the sample is small, we cannot apply the CLT. • So, we examine the sample data • Since the sample data is roughly normal, the sampling distribution will be approximately normal.

  12. Example: Runners • Step 3:If the conditions are met, Namethe inferenceprocedure, state theTeststatistic, andObtainthe p-value: • We will perform a 1-sample t-test

  13. Example: Runners • Step 4: Make a decision (reject or fail to reject H0). State your conclusion in context of the problem using p-value. • The p-value is small enough, p-value = 0.0171, that we reject the null hypothesis in favor of the alternative at a 95%confidence level. There is sufficient evidence to say that the mean weight of the runners is less than 64kg. This implies that the diet worked. • However, our independence assumption may have been violated. If so, our results may not be valid.

  14. Assignment

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