Question

A marketing analyst wants to determine how much higher the proportion of graduate students who drink coffee is compared to the proportion of undergraduate college students that drink coffee.

He selects a simple random sample of 55 graduate students and 54 undergraduate students and finds that 37 of the undergraduate students and 33 of the graduate students are coffee drinkers.

Answer 1

The proportion of graduate students who drink coffee is not higher than the proportion of undergraduate college students that drink coffee.

Explanation:

In this case we need to determine if the proportion of graduate students who drink coffee is higher than the proportion of undergraduate college students that drink coffee.

The hypothesis can be defined as follows:

H₀: The proportion of graduate students who drink coffee is not higher than the proportion of undergraduate college students that drink coffee, i.e. p₁ - p₂ ≤ 0.

Hₐ: The proportion of graduate students who drink coffee is higher than the proportion of undergraduate college students that drink coffee, i.e. p₁ - p₂ > 0.

The test statistic is defined as follows:

`z=\frac{\hat p_(1)-\hat p_(2)}{\sqrt{\hat P(1-\hat P)[(1)/(n_(1))+(1)/(n_(2))]}}`

The information provided is:

n₁ = 55

n₂ = 54

X₁ = 33

X₂ = 37

Compute the sample proportions and total proportions as follows:

`\hat p_(1)=(X_(1))/(n_(1))=(33)/(55)=0.60`

`\hat p_(2)=(X_(2))/(x_(2))=(37)/(54)=0.69`

`\hat P=(n_(1)X_(1)+n_(2)X_(2))/(n_(1)+n_(2))=(33+37)/(55+54)=0.64`

Compute the test statistic value as follows:

`z=\frac{\hat p_(1)-\hat p_(2)}{\sqrt{\hat P(1-\hat P)[(1)/(n_(1))+(1)/(n_(2))]}}`

`=\frac{0.60-0.69}{\sqrt{0.64(1-0.64)[(1)/(55)+(1)/(54)]}}`

`=-0.98`

The test statistic value is -0.98.

The decision rule is:

The null hypothesis will be rejected if the p-value of the test is less than the significance level.

Compute the p-value as follows:

`p-value=P(Z>-0.98)\\=1-P(Z<0.98)\\=1-0.83646\\\approx0.1635`

The p-value of the test is quite large. The null hypothesis will not be rejected at any significance level.

Thus, the proportion of graduate students who drink coffee is not higher than the proportion of undergraduate college students that drink coffee.