Question

In Exercises 7.1 to \(7.4,\) find the expected counts in each category using the given sample size and null hypothesis. $$ \begin{aligned} &\text { 7.4 } H_{0}: p_{1}=0.7, p_{2}=0.1, p_{3}=0.1, p_{4}=0.1 ;\\\ &n=400 \end{aligned} $$

Short answer

The expected counts for the categories are: \(E_1=280\), \(E_2= 40\), \(E_3= 40\) and \(E_4= 40\).

Step-by-step solution

Understand the proportions given for each category

The null hypothesis \(H_{0}\) states that the proportions of categories are \(p_{1} = 0.7,\) \(p_{2} = 0.1,\) \(p_{3} = 0.1,\) and \(p_{4} = 0.1.\)

Calculate expected counts for each category

The expected counts, denoted as \(E\), for each category can be calculated using the formula \(E = np\), where \(n\) is the total sample size and \(p\) is the hypothesised proportion for the category. So, the expected counts for each category are: \[E_{1} = np_{1} = 400*0.7 = 280\] \[E_{2} = np_{2} = 400*0.1 = 40\] \[E_{3} = np_{3} = 400*0.1 = 40\] \[E_{4} = np_{4} = 400*0.1 = 40\]