Hypothtesis Test

Please see attachment and explain all work.

#1
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.

#2
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?