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Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
When conducting a hypothesis test with a binomial distribution (sometimes called a Binomial Test), there are three ways to calculate the P-value (with additional variations possible). The only exact calculation is to use the binomial probability distribution. The other methods are approximations using the standardized
This problem introduces the method to obtain an exact P-value using the binomial distribution. Also, this method uses Excel to obtain the answers.
For this demonstration problem, we will test a hypothesis that a population proportion has increased since the last time it was measured. Previously, the population proportion was measured at 20%. For the current analysis, a sample of n=150n=150 randomly chosen subjects was obtained, and 37 of those demonstrated the observation of interest (i.e., a success).
To start, we clearly construct the hypotheses for this problem. Because the researcher suggested the proportion has increased, this would suggest a one-tailed test (as can be seen in the choice of HaHa):
Ho:p=0.2Ho:p=0.2
Ha:p>0.2Ha:p>0.2
Using the binomial distribution, the test statistic from the sample would simply be the sample count (the number of successful observations):
k=37k=37
The P-value for this scenario would be observing this count or one more extreme. With the alternative hypothesis suggesting that values at or below 20% would be unsurprising, this would suggest that the observed count or larger would constitute the potentially extreme responses. Thus, the P-value would be:
P(X≥37∣p=0.2,n=150)P(X≥37∣p=0.2,n=150)
Using Excel, this value can be calculated exactly:=1-BINOMDIST(36,150,20%,TRUE)which should return a P-value of 0.0945. Thus, with a traditional significance level of either α=0.05α=0.05 or α=0.01α=0.01, this P-value would result in failing to reject the null hypothesis. Thus, there is not enough sample evidence to support the claim that the population proportion has increased.
Note: For a two-tailed hypothesis test, the calculation can become a bit tricky. In this case, it is necessary to take the counts at and above the sample count, but it is also necessary to determine comparable counts below the hypothesized population count. This would be obtained using the formulan⋅p−(k−n⋅p)n⋅p-(k-n⋅p)This value is the count below the hypothesized mean count by the same distance as the sample count was above the mean count. For this demonstration example, this value would be 23. To obtain the P-value from Excel, you would use the following formula:=BINOMDIST(23,150,20%,TRUE)+1-BINOMDIST(36,150,20%,TRUE)Though, it is recommended that the two-tailed test only be used with one of the approximation methods as the calculations are less cumbersome.
Exercise Problem
In 1977, only 10% of the students in the city school district were classified as being learning disabled. A school psychologist suspects that the proportion of learning-disabled children has increased dramatically over the years. To demonstrate this point, a random sample of n=200n=200 students is selected. In this sample there are 27 students who have been identified as learning-disabled. You will use this information to determine if the sample indicates a change in the proportion of learning-disabled students at a 0.02 level of significance.
What is the hypothesized (assumed constant) population proportion for this test?
p=p=
(Report answer as a decimal accurate to 2 decimal places. Do not report using the percent symbol.)
Based on the researcher's understanding of the situation, how many tails would this hypothesis test have?
- one-tailed test
- two-tailed test
Choose the correct pair of hypotheses for this situation:
| (A) | (B) | (C) |
|---|---|---|
| H0:p=0.1H0:p=0.1 Ha:p<0.1Ha:p<0.1 |
H0:p=0.1H0:p=0.1 Ha:p≠0.1Ha:p≠0.1 |
H0:p=0.1H0:p=0.1 Ha:p>0.1Ha:p>0.1 |
| (D) | (E) | (F) |
| H0:p=0.135H0:p=0.135 Ha:p<0.135Ha:p<0.135 |
H0:p=0.135H0:p=0.135 Ha:p≠0.135Ha:p≠0.135 |
H0:p=0.135H0:p=0.135 Ha:p>0.135Ha:p>0.135 |
The test statistic for this analysis is the sample count (i.e., the number of observed successes). What is this value?
k=
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