(Due: May 26 Quiz)
Explain your answers and show your work clearly
1. Concept questions about hypothesis testing (9ts).
a) How is ‘the standard deviation of a sample’ different from ‘standard error of sampling distribution’? Please explain (3pts)
While ‘the standard deviation of a sample’ is a standard deviation of one sample, ‘standard error of sampling distribution’ is 1) calculated from the sample means of the same sample size and 2) can be similar to population standard deviation divided by the square root of sample size, (this part forward is optional) as long as the samples are randomly taken and sample size is large enough.
b) Why do we say “Do not reject Ho”, instead of “Accept Ho”? Please explain (2pts.)
A test starts by assuming Ho is true. If the sample evidence is not enough to reject Ho then the test is inconclusive. When we do not reject Ho, it means we do not have enough statistical evidence to say that Ho is false.
c) When do you choose left-tail test in one population proportion test? (2pts.)
When we want to test if one population proportion is less than a certain claimed value (proportion)
d) What is ‘p-value’ and how can we find a p-value for two-tail Z-test for population mean? (2pts.)
P-value is the tail probability of z-value(s) from test statistics, calculated from claimed means and sample means with test statistics for population proportion formula. To get p-value for 2 tail z-test, we should first get Z-test statistics from formula and find the tail probability of the z-test stat from z-table. Then, we should multiply it by 2 because the test is two-tail test.
2. The quality control manager at a light bulb factory took a sample of 64 light bulbs and found that the sample mean life is 350 hours. The population standard deviation is 100 hours.
a) Test at the 5% level of significance whether the mean life of light bulbs is different from 375 hours. Use the p-value method.
H0: µ = 375 H1: µ ≠ 375 (it implies 2-tail test)
Z test (Population STDEV known as 100), α = 0.05
Z test = (350-375) / (100/sqrt(64)) = -2.0
P-value = 2*(0.5-0.4772) = 0.0456, p-value <alpha
We reject H0
At the 5% level of significance, we found evidence to conclude that the mean life of light bulbs is different from 375 hours.
b) Test at the 2% level of significance whether the mean life of light bulbs is less than 370 hours. Use the rejection-region method.
H0: µ ≥ 370 H1: µ < 370 (it implies left-tail test)
Z test (Population STDEV known as 100), α = 0.02
Z critical = -2.055 (2.05, 2.06 works)
Z test = (350-370) / (100/sqrt(64)) = -1.6
Z-test > Z-critical (left tail) We do not reject H0
At the 2% level of significance, we did not find evidence that the mean life of light bulbs is less than 370 hours.
3. Hypothesis test questions (8pts)
a) The US department of education reports that 40% of full time college students are employed while attending college. A recent survey of 60 full time students at a university found that 28 are employed. Please test whether the proportion of full time students who are employed while attending the university is different from the national norm of 40%. Please use p-value method with alpha = 0.01.
H0: π = 0.4 H1: π ≠ 0.40 (it implies 2-tail test.)
Z test, α = 0.01
Point Estimate of Proportion = 28/60 = 0.4667
Z-test = (0.4667 - 0.4) / sqrt(0.4*0.6/60) = 1.0546 we use 1.05 in Z table
P-value = 2*(0.5- 0.3531) = 0.2938
P-value > alpha, we do not reject H0
At 1% level of significance, we did not find evidence that the proportion of full time students who are employed while attending the university is different from the national norm of 40%.
b) The worldwide market share for the Mozilla Firefox browser was 18.35% in recent month. Suppose you selected a sample of 100 students at your university and found that 24 use the Mozilla Firefox. Test the hypothesis at the 0.05 level whether the market share for the Firefox browser at your