In: Statistics and Probability

**A study published in 2009 (Goldstein et al, 2009) asked
a random selection of 499 adults from various regions in Israel to
complete a survey about whether or not they read while using the
toilet. The researchers conducted a test of Ho: pi = 0.5 versus Ha:
pi ≠ 0.5 where pi is the proportion of adults who read while using
the toilet.**

- Use a theory-based method to find the rejection region of the test using a significance level of alpha = 0.1 and p-hat as the test statistic. Recall, the rejection region is the set of values of p-hat that would lead the researchers to reject the null hypothesis.
- Suppose the true proportion of adults who read while sitting on the toilet is 50%. What is the probability the researchers will incorrectly conclude that the proportion is different from 50%?
- Suppose the true proportion of adults who read while sitting on the toilet is 55%. Use a theory-based method to calculate the probability the test will reveal that the alternative hypothesis is true.
- Suppose the researchers were able to collect more data from another 100 adults, so they had a total of 599 adults in their sample. What would happen to the power of the test; would it increase, decrease, or stay the same? Explain.

**ANSWER::**

(a)

The rejection region:

if phat <= 0.4632 or phat >= 0.5368 , reject H0

(b)

Here we need to find the probability that researcher incorrectly reject the true null hypothesis.

It is type I error. It must be equal to 0.10.

(c)

Here we need to calculate the power of the test. The rejetion region we need to use in of part a.

(d)

The power for n= 599:

That is power decreased for n=599.

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A study asked a random selection of 499 adults from various
regions in Israel to complete a survey about whether or not they
read while using the toilet. The researchers conducted a test of
H0: pi = 0.5 versus Ha: pi ≠ 0.5 where pi is the proportion of
adults who read while using the toilet.
a. Use a theory-based method to find the rejection region of the
test using a significance level of a = 0.1 and phat as...

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