Test Hypotheses with sample of Binomial RV's

[LBJking123]

Hi, I am trying to teach myself how to test hypotheses for any distribution, but am having some trouble.

X=number chosen each year
θ=Mean number chosen in the population

H0: θ=.5
h1: θ>.5

The random sample of n=4 is 0,1,3,3

Test the Hypotheses at α≤0.05 assuming X is a binomial(5,θ/5).

This is what I have so far, but I feel I am completely missing something..

Sample average (Xbar = 1.75

So,

Reject H0 if P(Xbar≥1.75, given that X is binomial(5,.1)) ≤ 0.05

Then I figure out 1-P(Xbar≤1.75)=0.08146 which is greater than 0.05 so I reject the null.

I know something is not right... Any help would be much appreciated. Thanks!

 

[chiro]

Hey LBJking123.

I'm a little busy at the moment, but I think I found a document that will help you:

http://www.google.com.au/url?sa=t&r...eB8IBw&usg=AFQjCNEKV-v9v-rGZgYzsitOmWHyxBepaA

 

[LBJking123]

So would I want to do a z score, like on page three of that document? That seems like it would work better.

Then, I get P(Z≥(1.75-.5)/SQRT(.25/4))=P(Z≥5)=0, which is less than 0.05 so I reject the null..?

 

[chiro]

If that distribution and region corresponds to H0 then yes you reject the null.

Remember that a p-value is looking at a probability for some estimator distribution that relates to a hypothesis.

 

[blue_raver22]

Dear researcher,

Your approach is on the right track, but there are a few errors in your calculations. Here is a step-by-step explanation of how to properly test the hypotheses using a sample of binomial random variables:

1. Define the null and alternative hypotheses:
- H0: θ = 0.5
- H1: θ > 0.5

2. Calculate the test statistic:
The test statistic for this situation is the sample proportion, which is calculated by dividing the sum of the sample (X) by the sample size (n). In this case, the sample proportion is 1.75/4 = 0.4375.

3. Determine the critical value:
Since we are testing at α = 0.05, we need to find the critical value for a one-tailed test with α = 0.05. This can be found using a table of critical values for the binomial distribution. In this case, the critical value is 3, meaning that we will reject the null hypothesis if the sample proportion is greater than 3.

4. Calculate the p-value:
The p-value is the probability of obtaining a sample proportion as extreme or more extreme than the one observed, assuming the null hypothesis is true. In this case, we are looking for the probability of obtaining a sample proportion greater than or equal to 0.4375 in a binomial distribution with n = 4 and θ = 0.5. This can be calculated using a binomial probability calculator or a table of binomial probabilities. In this case, the p-value is 0.082.

5. Compare the p-value to the significance level:
Since the p-value (0.082) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that θ is greater than 0.5.

In summary, your approach was correct but there were some errors in the calculations. It is important to carefully follow the steps for hypothesis testing to ensure accurate results. Keep practicing and seeking help when needed, and you will become more confident in testing hypotheses for any distribution. Good luck with your research!