Question

An FM rock station claims that 85% of all its listeners are college students. A statistician decides to test this claim against the suspicion that the percentage is too high.
A random sample of 2000 listeners is selected from the population of all listeners and it is determined that 1655 are college students. Perform a hypothesis test to answer the question:

Do the sample results support the statistician’s claim at α = 1 %?

3 I. State the hypotheses (in words & symbols):

H0: _______________________________________________________________________________ _______________________________________________________________________________

Ha: _______________________________________________________________________________ _________________________________________________________________________________

II. Determine the model to test H0
Can one use a normal distribution for the Sampling Distribution model to perform this test? Please explain. ________________________________________________________________________

_________________________________________________________________________________

b) Using the appropriate notation, state the MEAN & STD ERROR of the sampling distribution:

Mean: _______________________________________________________________________
STD Error: ____________________________________________________________________

III. Determine the Decision Rule:

State the decision rule: ________________________________________________________

____________________________________________________________________________

IV. Analyze the sample data: Using the appropriate notation, state the sample result of the test, p-value and the test statistic.

sample result is: ____________________________________________________

p-value is: _________________________________________________________

test statistic is: _____________________________________________________

V. State the Conclusion: ___________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________

4Can the statistician reject the rock station's claim at α = 1%? YOU MUST EXPLAIN YOUR ANSWER OTHERWISE: No credit will be given.

_________________________________________________________________________________

_________________________________________________________________________________

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.85
Alternative Hypothesis, Ha: p < 0.85

normal distribution can be used
because np > 5 and nq > 5
n = 2000 and p = 0.85 and q = 1 - 0.85

pcap = 1655/200 = 0.8275

SE = sqrt(0.85*(1-0.85)/2000) = 0.008

III)
This is left tailed test, for α = 0.01
Critical value of z is -2.33.
Hence reject H0 if z < -2.33

IV)
Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.8275 - 0.85)/sqrt(0.85*(1-0.85)/2000)
z = -2.82

P-value = 0.0024
As P-value < 0.01, reject the null hypothesis.

There is statistical significance that the proportion of 0.85 is too high