Question

“Will changing the rating scale on a survey affect how people answer the question?” To find out, the group took an SRS of $50$students from an alphabetical roster of the school’s just over $1000$students. The first $22$students chosen were asked to rate the cafeteria food on a scale of $1$(terrible) to $5$(excellent). The remaining $28$students were asked to rate the cafeteria food on a scale of $0$(terrible) to $4$(excellent). Here are the data:

The students decided to compare the average ratings of the cafeteria food on the two scales.

a. Find the mean and standard deviation of the ratings for the students who were given the $1to5$scale.

b. For the students who were given the $0to4$scale, the ratings have a mean of $3.21$and a standard deviation of $0.568$. Since the scales differ by one point, the group decided to add $1$to each of these ratings. What are the mean and standard deviation of the adjusted ratings?

c. Would it be appropriate to compare the means from parts (a) and (b) using a two-sample t test? Justify your answer

Short answer

Part a. $Mean=3.5455$and standard deviation$=1.1843$

Part b. The adjusted mean$=4.21$and adjusted $sd=0.568$

Part c. No.

Step-by-step solution

Part a. Step 1. Explanation

The mean is,

$\overline{x}=\frac{1\times 2+2\times 3+3\times 1+4\times 13+5\times 3}{2+3+1+13+3}=3.5455$

The standard deviation,

$$\begin{gathered} s=\sqrt{\frac{{(1-3.5455)}^2\times 2+{(2-3.5455)}^2\times 3+{(3-1.3455)}^2\times 1+{(4-3.5455)}^2\times 13+{(5-3.5455)}^2\times 3}{2+3+1+13+3-1}} \\ s=1.1843 \end{gathered}$$

Part b. Step 1. Given information

Mean and standard deviation for scale 0-4:

$$\begin{gathered} {\overline{x}}_{0-4}=3.21 \\ {\overline{s}}_{0-4}=0.568 \end{gathered}$$

Part b. Step 2. Explanation

The adjusted mean and standard deviation are,

$$\begin{gathered} {\overline{x}}_{(0-4)+1}={\overline{x}}_{0-4}+1=3.21+1=4.21 \\ {\overline{s}}_{(0-4)+1}=0.568 \end{gathered}$$

Part c. Step 1. Explanation

The given variables having ratings with corresponding to their frequency. Therefore, these are categorical variables. For two sample t test, data should be numerical. Hence, it is not appropriate to use a two-sample t test.