Question

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A random sample of 1,000 adult U.S. citizens were surveyed; 210 of them indicated that playing the lottery would be the most practical way for them to accumulate $200,000 in net wealth in their lifetime.

A journalist reporting this story asks one of their statistician friends if this survey provides convincing evidence that more than 20% of adult U.S. citizens believe that playing the lottery is the best strategy for accumulating such wealth.

This particular statistician isn’t a helpful person, generally. They reply, “A hypothesis test of a proportion gives the p-value 0.215.”

a. What does the p-value mean in context?

b. What conclusion should the journalist report?

c. ​​​​​​​If the sample data led to an incorrect conclusion, which type of error occurred?

Answer 1

Step 1:

Ho: p ≤ 0.20

Ha: p > 0.20

Step 2:

n = 1000

x = 210

$z = \frac{x - np}{\sqrt{np(1-p)}} =\frac{ 210 - 1000 * 0.20}{\sqrt{1000 * 0.20(1-0.20)}}$

z = 0.791

z critical = 1.960

As z stat does not fall in rejection area, we fail to reject the Null hypothesis.

p value P(z > 0.791) = 0.2148

(a) P values indicate whether hypothesis tests are statistically significant. P value is the probability of getting observed or more extreme result ( more than 20% of adult U.S. citizens believe that playing the lottery is the best strategy for accumulating such wealth ) assuming that the null hypotheiss (i.e. p ≤ 0.20) is true.

(b) Assuming level of significance at 0.05

As p value is greater than alpha, we fail to reject the Null hypothesis.

Hence we do not have sufficient evidence to beileve that  more than 20% of adult U.S. citizens believe that playing the lottery is the best strategy for accumulating such wealth

(c)

Accept/reject	Ho True	Ho False
Reject Ho	Type I error	correct
Accept Ho	correct	Type II error

Chances of making Type II error, i.e. the Ho is false but as per the sample data we fail to reject the Null hypothesis.