Let p equal the proportion of yellow candies in a package

Chapter 8, Problem 12E

(choose chapter or problem)

Let p equal the proportion of yellow candies in a package of mixed colors. It is claimed that \(p = 0.20\).

(a) Define a test statistic and critical region with a significance level of \(\alpha=0.025\) for testing \(H_{0}: p=0.20\) against a two-sided alternative hypothesis.

(b) To perform the test, each of 20 students counted the number of yellow candies, \(y\), and the total number of candies, \(n\), in a \(48.1\)-gram package, yielding the following ratios, \(y / n: 8 / 56,13 / 55,12 / 58,13 / 56,14 / 57,5 / 54\), \(14 / 56,15 / 57,11 / 54,13 / 55,10 / 57,8 / 59,10 / 54,11 / 55\), \(12 / 56,11 / 57,6 / 54,7 / 58,12 / 58,14 / 58\). If each individual tests \(H_{0}: p=0.20\), what proportion of the students rejected the null hypothesis?

(c) If we may assume that the null hypothesis is true, what proportion of the students would you have expected to reject the null hypothesis?

(d) For each of the 20 ratios in part (b), a \(95 \%\) confidence interval for \(p\) can be calculated. What proportion of these \(95 \%\) confidence intervals contain \(p=0.20\)?

(e) If the 20 results are pooled so that \(\sum_{i=1}^{20} y_{i}\) equals the number of yellow candies and \(\sum_{i=1}^{20} n_{i}\) equals the total sample size, do we reject \(H_{0}: p=0.20\)?

Equation Transcription:

   

, .

 

 

Text Transcription:

y  

n  

48.1    

y/n:8/56,13/55,12/58,13/56,14/57,5/5414/56,15/57,11/54,13/55,10/57,8/59,10/54,11/55, 12/56,11/57,6/54,7/58,12/58,14/58

H_0:p=0.20  

95%  

p  

p=0.20  

sum_i=1^20 y_i  

sum_i=1^20 n_i