7. In 2015, the Census Bu reported that 20% all Maryland households had income a

Question

7. In 2015, the Census Bu reported that 20% all Maryland households had income above During that same of400 selected Montgomery County households, 100 had year, in a survey Do the data provide sufficient evidence is greater that the proportion above S100,000. County households with income above S100,000 of all Montgomery sing than .20 the proportion households with income above s100,000. provides proper notation define the parameter and state the hypotheses for testing if this sample 100,000 evidence that the proportion of Montgomery County households with income above is greater than 0.20, the proportion of all Maryland h with income above 7, 20 b. Calculate the observed sample statistic. Use proper notation. c. Describe how you would generate a single randomization sample in this situation, and identify the statistic you would record for each sample. d. Use the provided randomization distribution based on 100 samples) to estimate the p-value for this sample. 0.14 0.18

Explanation / Answer

a. Null Hypothesis H0: The proportion of Montgomery County Households with income above $100,000 is equal to 0.20, p = 0.20

Alternative Hypothesis H1: The proportion of Montgomery County Households with income above $100,000 is greater than 0.20, p > 0.20

b. proportion of Montgomery County Households with income above $100,000 = 100/400 = 0.25

Standard error = sqrt(p(1-p)/n) = sqrt(0.25*0.75/400) = 0.0216

sample statistic = (0.25 - 0.20) / 0.0216 = 2.315

c. Randomly select the Montgomery County Households and calculate the proportion of Households with income above $100,000 (statistic) of the random sample.

d. mean = 0.199

standard error = 0.020

z-value for 95% confidence interval (5% significance level) is 1.64

For right tail, test statistic = (x - 0.20)/ 0.020 = 1.64

x = 0.2328

Points to the right of x = 0.2328 in the graph are 10. So p-value for right tail is 10/100 = 0.1

Left tail - test statistic = (x - 0.20)/ 0.020 = -1.64

x = 0.1672

Points to the left of x = 0.1672 in the graph are 2. So p-value for right tail is 2/100 = 0.02

Two-tail - p-value of 2-tail is sum of p-values of left and right tail = 0.1 + 0.02 = 0.12

e. As in (b), p =0.25, which lies to the right of 95% confidence interval (p=0.1), so we reject the null hypothesis and accept the alternative hypothesis that the proportion of Montgomery County Households with income above $100,000 is greater than 0.20. Yes the result are statistically significant.

f. Answer is A - Type I error. We have rejected the null hypothesis, but in fact the null hypothesis is true.

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