Q.63 - The Practice of Statistics for AP

Question

Average ratings  $(1.3,10.2)$ 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 $1$ -to-$5$ scale.

(b) For the students who were given the $1$-to-$5$ 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 

Step-by-Step Solution

Answer

a). $\overline{x}=3.545$,

$s=1.1857$

b). Adjusted Mean is $4.21$,

Adjusted standard deviation is $0.568$.

c). Two sample test for mean isn't appropriate.

Part (a) Step 1: Given Information

Total number of people is $90$.

The table of the frequency count of the ratings $1$ to $5$ is as follows

Part (a) Step 2: Explanation

Calculation

$x$	$f$	$xf$	$x^{2}f$
1	2	2	2
2	3	6	12
3	1	3	9
4	13	52	208
5	3	15	75
	$\sum f=22$	$\sum (x\times f)=78$	$\sum \left(x^2\times f\right)=306$

The mean will be

$\overline{x}=\frac{78}{22}$

$=3.545$

The standard deviation will be

$s=\sqrt{\frac{1}{22-1}\times \left(306-22\times 3.{545}^2\right)}$

$=1.1857$

Part (b) Step 1: Given Information

Mean of the ratings $0$ to $4=3.21$.

The standard deviation of the ratings $0$ to $4=0.568$.

Part (b) Step 2: Explanation

Concept used:

$E[aX+b]=a\times E\left[X\right]+b$

$Sd(aX+b)=a\times Sd\left(x\right)$

Using the above concept as an addition or subtraction in all the terms leads to the simultaneous changes in the mean while the standard deviation is affected by only multiplication or division of all the terms.

As $1$ is added to each of the ratings, so

The mean of the adjusted ratings will be${\overline{x}}^{\text{'}}=\overline{x}+1$

                                                                       $=3.21+1$

                                                                       $=4.21$

The adjusted standard deviation will remain the same as it remains unaffected by the addition or subtraction of the value.

Part (c) Step 1: Given Information

To compare the means from parts (a) and (b) using a two-sample t-test. 

Part (c) Step 2: Explanation

Two sample test for means is used when the data is nominal in nature.

In this case, the ratings are to be tested an ordinal data (ordered data)

Since the data is categorical in nature, so two-sample $t$-test is not appropriate.