problem set 4 | Mathematics homework help

2.       Fill in the blanks. A particular random sample of n observations can be used to calculate a sample proportion. The count of successes in the sample will vary according to a (a) ____________________ probability distribution with parameters n and (b) ____________________. When the sample is large, the number of success will vary according to a normal distribution with μ = (c) ____________________ and σ = (d) ____________________. At the same time, the sampling distribution of proportion in large samples will be Normally distributed with mean p with standard deviation equal to (e) ____________________.

3.       Misconceived hypotheses. What is wrong with each of the following hypothesis statements?

a)      H₀: μ = 100 vs. H_a: μ ≠ 110

b)      H₀: = 100 vs. H_a: < 100

c)       H₀: = 0.50 vs. H_a: ≠ 0.50

4.       Patient satisfaction. Scores derived from a patient satisfaction survey are Normally distributed with μ = 50 and σ = 7.5, with high scores indicating high satisfaction. An SRS of n = 36 is taken from this population.

a)      What is the SE of x̅ for these data?

b)      We seek evidence against the hypothesis that a particular group of patients comes from a population in which μ = 50. Sketch the curve that describes the sampling distribution of x̅ under the null hypothesis. Mark the horizontal axis with values that are ±1, ±2, and ±3 standard errors above and below the mean.

c)       Suppose a sample of n = 36 finds an x̅ of 48.8. Mark this finding on the horizontal axis of your sketch. Then calculate the z_stat for the result. Does this observation provide strong evidence against H₀?

5.       Patient satisfaction (sample mean of 48.8). In Exercise 4, you tested H₀: μ = 50 based on a sample of n = 36 showing = 48.8. The population had standard deviation σ = 7.5.

a)      What is the one-sided alternative hypothesis for this test?

b)      Calculate the z-statistic for the test.

c)       Convert the z_stat to a P-value, and interpret the results.