Question

ten young adults living in california rated the taste of a newly developed sushi pizza topped with tuna, rice, and kelp on a scale of 1 to 50, with 1 indicating they did not like the taste and 50 that they did. the ratings were: 34 39 40 46 33 31 34 14 15 45 in a parallel study, 10 young adults in iowa rated the taste of the same pizza. the ratings were: 28 25 35 16 25 29 24 26 17 20 as a market researcher, compare the potential for sushi pizza in the two markets. click here for the excel data file required: a. calculate the range, mean, and variance. note: round \mean\ to 1 decimal place and \variance\ to 2 decimal places.

Explanation:

Step1: Calculate range for California

Range = Max - Min. For California ratings (34, 39, 40, 46, 33, 31, 34, 14, 15, 45), Max = 46, Min = 14. So range = 46 - 14 = 32.

Step2: Calculate mean for California

Mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. $\sum_{i=1}^{10}x_{i}=34 + 39+40 + 46+33+31+34+14+15+45=331$. $n = 10$. Mean = $\frac{331}{10}=33.1$.

Step3: Calculate variance for California

Variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$.
$(34 - 33.1)^{2}=0.81$, $(39 - 33.1)^{2}=34.81$, $(40 - 33.1)^{2}=47.61$, $(46 - 33.1)^{2}=166.41$, $(33 - 33.1)^{2}=0.01$, $(31 - 33.1)^{2}=4.41$, $(34 - 33.1)^{2}=0.81$, $(14 - 33.1)^{2}=364.81$, $(15 - 33.1)^{2}=327.61$, $(45 - 33.1)^{2}=141.61$.
$\sum_{i = 1}^{10}(x_{i}-33.1)^{2}=0.81+34.81+47.61+166.41+0.01+4.41+0.81+364.81+327.61+141.61 = 1088.9$.
Variance = $\frac{1088.9}{9}\approx120.99$.

Step4: Calculate range for Iowa

For Iowa ratings (28, 25, 35, 16, 25, 29, 24, 26, 17, 20), Max = 35, Min = 16. Range = 35 - 16 = 19.

Step5: Calculate mean for Iowa

$\sum_{i=1}^{10}x_{i}=28+25+35+16+25+29+24+26+17+20 = 245$. Mean = $\frac{245}{10}=24.5$.

Step6: Calculate variance for Iowa

$(28 - 24.5)^{2}=12.25$, $(25 - 24.5)^{2}=0.25$, $(35 - 24.5)^{2}=110.25$, $(16 - 24.5)^{2}=72.25$, $(25 - 24.5)^{2}=0.25$, $(29 - 24.5)^{2}=20.25$, $(24 - 24.5)^{2}=0.25$, $(26 - 24.5)^{2}=2.25$, $(17 - 24.5)^{2}=56.25$, $(20 - 24.5)^{2}=20.25$.
$\sum_{i = 1}^{10}(x_{i}-24.5)^{2}=12.25+0.25+110.25+72.25+0.25+20.25+0.25+2.25+56.25+20.25 = 294.5$.
Variance = $\frac{294.5}{9}\approx32.72$.

Answer:

	Range	Mean	Variance
Iowa	19	24.5	32.72