Another Statistics /Probability question!!?
November 9th, 2009 | 
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There is speculation that 20% of students will have laptops. If more than 20% of people own laptops, a college will need to provide more space for laptop use. Describe in detail, the hypothesis test you would use to test this? Include how you would sample, how you would test and what conclusions could be drawn from the results.
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Start by finding the null and alternate hypothesis.
H0: p ≤ 0.20 vs. H1: p > 0.20
This works because if you reject the null you will conclude that the alternate H1 is true and the proportion is greater than 20%.
Note, if you have the set up as: H0: p ≥ 0.20 vs H1: p < 0.20 you can only conclude that the proportion is less than 20% or that it is plausible p is greater than 20%. not useful.
Hypothesis Test for proportions:
Let X be the number of success in n independent and residentially distributed Bernoulli trials, i.e., X ~ Binomial(n, p)
To test the null hypothesis of the for H0: p ≤ p0, H0: p ≥ p0 or H0: p = P0
assuming that n*p0 > 10 and n * (1-p0) > 10 then
find the test statistic z = (pHat – p0) / sqrt(p0 * (1-p0) / n)
where pHat is the observed success divided by the sample size n
The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.
H1: p > p0; p-value is the area to the right of z
H1: p < p0; p-value is the area to the left of z
H1: p ≠ p0; p-value is the area in the tails greater than |z|
The claim is that "more than 20% of people will own laptops." That is:
p > .20
Because this statement does NOT contain equality, it must be the alternative hypothesis. The null hypothesis, depending on how your book/instructor has presented it, will be either:
p = .20 OR p ≤ .20
Now you need sample statistics to test the null hypothesis. I’m not sure what exactly they’re looking for here, other than to say one should randomly sample at least 30 students to ask them if they have a laptop.
The number of students who answer that they do have a laptop is x (the number of successes) and the number of students you ask is n (the number of trials). p-hat is the sample proportion, equal to x over n.
Now you need to calculate the probability of getting a sample proportion of your p-hat OR GREATER. You can do this by using a normal curve to approximate the binomial. The normal curve will have a mean of the assumed p, which is .20. The standard deviation will be the square root of pq / n , where p is still .20, q is the complement, or .80, and n is still the number of people you sampled.
That calculation of the probability of getting your sampled p-hat or greater is called the p-value. If this p-value is found to be small enough (defined by whomever is interested in these results, the hypothetical college, maybe? and it’s called the significance level and is abbreviated α) then the assumption of p = .20 is concluded to be false. We say we "Reject the null hypothesis". Rejection of the null hypothesis, in this case, supports the alternative hypothesis. This is the claim that more than 20% of the students have laptops. We would say that your collected data is sufficient to support the claim that more than 20% of the students have laptops.
If the p-value is NOT small enough ( p-value > α ), then we "fail to reject the null hypothesis," and conclude that your sampled data is not sufficient to support the claim that 20% of the students own laptops.