Statistics Question

Question

Suppose Brian believes that over 50% of the packages include more than 140 grams of chocolate. Based on the same sample of 16 packages, he is interested in testing his belief. Write down the formal null and alternative hypotheses she should test for this purpose, conduct the hypothesis test, and state your conclusion in terms appropriate for the context 

Answer 1

Hi Chris,

First things first. When stating statistical hypotheses, null hypothesis (H₀) is always that, nothing. In this case, no difference. Alternative hypothesis (H_A) is always something, some difference. In this case, the problem specified more than 140g chocolate, so our alternative is greater than. So:

H₀: Packages contain 140g chocolate or under on average.

H_A: Packages contain over 140g chocolate on average.

Moving on to the test, we have only 16 bags and no known population standard deviation, so we can't assume normality for z. We have to use t. Now, our t-test statistic is:

t=(x-bar - mu₀/SE)

Breaking this down,

x-bar=sample mean

mu₀=value you are testing

SE=standard error

Specifically for this problem:

x-bar=143.61

mu₀=140

SE=13.191

Thus, we have:

t= (143.61-140)/13.191

t=0.274

Let's talk about degrees of freedom. In the one-sample t-test, this is just: df= n-1, so

df=16-1

df=15

Now, we need the t-critical value t*. Now, your problem did not specify a confidence level, but the usual standard is 95%, so go to the t-table, look for 95% confidence, one tail--remember we want greater than only--and 15 degrees of freedom. This gives:

t*=2.131

Our calculated t, 0.274, is far less than this, so we fail to reject the null hypothesis and conclude that package chocolate content does not exceed 140g on average. This is your conclusion. I hope this helps.