Question

It is believed that approximately 12% of the population of the United States is lefthanded. Suppose researchers suspect that the proportion of left-handed people is higher in certain states than the national average. The researchers conduct a sample of 200 randomly selected people in the state of Maine and find that 29 people in the sample are left-handed.

a. Write the null hypothesis and alternative hypothesis and define your parameter.
b. Show that the necessary conditions (Randomization Condition, 10% Condition, Sample Size Condition) are satisfied to perform a hypothesis test. Briefly explain how each condition is satisfied.
c. Perform the hypothesis test and find the P-value. (To show your work: Write down what values you are entering into the hypothesis testing calculator.)
d. Is there strong evidence that the left-handed rate in the state of Maine is higher than the national average? Briefly explain how you know.

Answer 1

Explanation:

a. Null hypothesis: P = p

Alternatives hypothesis: P =/ p

Where P is the hypothesized population proportion and p is the sample proportion

b. Performing a test of proportions

Randomization: the sample was randomly selected in the study

The population size is at least 20 times as big as the sample size.

The sample includes both successes and failures with 29 success and 171 failures.

c. To perform the hypothesis test: we have to find the standard deviation first

Sd = sqrt[ P * ( 1 - P ) / n ]

where P is the hypothesized value of population proportion, n is the sample size.

Sd = √[0.12*(1-0.12)/200]

Sd = √[0.12*(0.88/200]

Sd = √[0.12*(0.0044)]

Sd = √0.000528

Sd = 0.023

Then we can find the z score

z = (p - P) / σ where p = 29/200 = 0.145

z = (0.145-0.12)/ 0.023

z = 0.025/0.023

z = 1.09

Calculation the p value using 0.05 level of significance and a two waited test (p value calculator),

A p-value of 0.2757 which is greater than 0.05, thus we will fail to reject the null stating that there is not enough strong evidence that the left-handed rate in the state of Maine is higher than the national average.