Question

People of taste are supposed to prefer fresh-brewed coffee to the instant variety. On the other hand, perhaps many coffee drinkers just want their caffeine fix. A sceptic claims that only half of all coffee drinkers prefer fresh-brewed coffee. To test this claim, we ask a random sample of 50 coffee drinkers in a small city to take part in a study. Each person tastes two unmarked cups-one containing instant coffee and one containing fresh-brewed coffee-and says which he or she prefers. We find that 36 of the 50 choose fresh coffee.

(a) We presented the two cups to each coffee drinker in a random order, so that some people tasted the fresh coffee first, while others drank the instant coffee first. Why do you think we did this?

(b) Do these results give convincing evidence that coffee drinkers favour fresh-brewed over instant coffee? Carry out a significance test to help answer this question.

Short answer

a. People have taken the coffee in a random order

b. coffee drinkers favour fresh-brewed coffee

Step-by-step solution

Introduction

The null hypothesis is a typical statistical hypothesis which recommends that no statistical relationship and significance exists in a bunch of given single noticed variables, between two arrangements of noticed information and estimated peculiarities.

Explanation Part (a)

In random order, two cups were presented to the people. some people drank fresh coffee first and some drank instant coffee.

Hence people have taken the coffee in random order.

Explanation Part (b) (1)

The sample size is n =$50$

People that chose fresh coffee x = $36$

role="math" localid="1652930402772" $\stackrel{-}{p}=\frac{x}{n}=\frac{36}{50}=0.72$

Calculating the null and alternative hypotheses,

$\begin{array}{r}{H}_0:p=0.5\\ H_a:p>0.5\end{array}$

Random selection of people from a sample of $50$,

role="math" localid="1652930469566" $n{p}_0=50\times 0.5=25$

$\begin{array}{r}n{p}_0\ge 10\end{array}$

$n\left(1-p_0\right)=50\times (1-0.5)=25$

$n\left(1-p_0\right)\ge 10$

Explanation Part (b) (2)

using,

$Z=\frac{\stackrel{-}{p}-p_0}{\sqrt{\frac{P_0\left(1-p_0\right)}{n}}}$

$$\begin{gathered} Z=\frac{0.72-0.50}{\sqrt{\frac{0.5(1-0.5)}{50}}} \\ =\frac{0.22}{0.0707} \\ =3.11 \end{gathered}$$

The p-value is $3.11$

Now, considering $P(Z<3.11)=0.9991$

$$\begin{gathered} P(Z=3.11)=1-P(Z<3.11) \\ =1-0.9991=0.0009 \end{gathered}$$

Hence we can say that coffee drinkers favour fresh-brewed coffee from the above evidence.