Question

1. Bernard ran an experiment to test optimum power and time settings for microwave popcorn. His goal was to deliver popcorn with fewer than 11​% of the kernels left​ unpopped, on average. He determined that power 9 at 4 minutes was the best combination. To be sure that the method was​ successful, he popped 8 more bags of popcorn​ (selected at​ random) at this setting. All were of high​quality, with the percentages of unpopped kernels shown below.

3.3, 7.5, 11.3, 2.3, 4.6, 13.1, 8.6, 7.1

Does this provide evidence that he met his goal of an average of fewer than 11​% unpopped​ kernels? Use 0.05 as the​ P-value cutoff level.

Calculate the test statistic. t =

Calculate the​ P-value.

Yes, there is enough evidence.

2. A company is criticized because only 18 of 43 executives at a local branch are women. The company explains that although this proportion is lower than it might​ wish, it's not a surprising value given that only 45​% of all its employees are women. The company has more than 500 executives worldwide. Test an appropriate hypothesis and state the conclusion.

Calculate the test statistic.

Find the​ P-value.

Answer 1

1. The null hypothesis: More than 11% of the kernels left unpopped i.e,  

The alternate hypothesis: Less than 11% of the kernels left unpopped i.e,  

Let X be the variable that denotes the percentage of unpopped kernels

From the given data we calculate mean and standard deviation

Test statistic;

degrees of freedom: d.f = n-1 = 7

{ p-value is the value of in the t-distribution table for which the value of    where and is the d.f }

Therefore, from the table we get the p-value = 0.013

Given level of significance = 0.05

p-value < 0.05

Therefore, we reject the null hypothesis and conclude that there is enough evidence that he met his goal of an average of fewer than 11% unpopped kernels.

2. The null hypothesis:

The alternete hypothesis;

Where, is the population proportion

sample proportion :  

Test statistic:

Let us take the significance level

{ p-value is the value of in the standard normal distribution table for which the value of   }

From the standard normal distribution table the p-value is calculated

p-value = 0.679

p-value > 0.05

Conclusion: We fail to reject the null hypothesis and conclude that although the proportion of women in local branch is low but it is not surprising.