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Hypothtesis Test

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Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that results in "acid rain." The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has a pH of 7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below 5.0. Suppose that pH measurements of rainfall on different days in a Canadian forest follow a Normal distribution with standard deviation σ = 0.5. A sample of n days finds that the mean pH is x̄ = 4.8. Is this good evidence that the mean pH μ for all rainy days is less than 5.0? The answer depends on the size of the sample.

Either by hand or using the P-Value of a Test of Significance applet, carry out three tests of:

H₀: μ = 5.0
Ha: &#956; < 5.0

Use &#963; = 0.5 and x&#772; = 4.8 in all three tests. But use three different sample sizes, n = 5, n = 15, and n = 40.

a) What are the P-values for the three tests? The P-value of the same result x&#772; = 4.8 gets smaller (more significant) as the sample size increases.

b) For each test, sketch the Normal curve for the sampling distribution of x&#772; when H&#8320; is true. This curve has a mean 5.0 and standard deviation 0.5/ &#8730;n. Make the observed x&#772; = 4.8 on each curve. (If you use the applet, you can just copy the curves displayed by the applet. The same result x&#772; = 4.8 gets more extreme on the sampling distribution as the sample size increases.

Suppose that scores of men aged 21 to 25 years on the quantitative part of the National Assessment of Educational Progress (NAEP) test follow a Normal distribution with standard deviation &#963; = 60. You want to estimate the mean score within ±10 with 90% confidence. How large an SRS of scores must you choose?


Solution Summary

Use pvalue to test the hypothesis.