the data to the right represent the number of chocolate chips per cookie in a random sample of a name brand and a store brand. complete parts (a) to (c) below.
(a) does there appear to be a difference in the number of chips per cookie?
a. yes. the store brand appears to have more chips per cookie.
b. no. there appears to be no difference in the number of chips per cookie.
c. yes. the name brand appears to have more chips per cookie.
d. there is insufficient information to draw a conclusion.
(c) does one brand have a more consistent number of chips per cookie?
a. yes. the name brand has a more consistent number of chips per cookie.
b. yes. the store brand has a more consistent number of chips per cookie.
c. no. both brands have roughly the same number of chips per cookie.
d. there is insufficient information to draw a conclusion.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the sum of chips for Name Brand
Name Brand data: $22,30,25,33,22,27,21,20,25,26,20,23,27,26,29,33,22$
Sum $S_{name}=\sum_{i = 1}^{17}x_{i}=22 + 30+25+33+22+27+21+20+25+26+20+23+27+26+29+33+22=439$
## Step2: Calculate the sum of chips for Store Brand
Store Brand data: $22,27,21,20,25,26,15,23,17,28,21,24,19,28,24$
Sum $S_{store}=\sum_{i = 1}^{15}y_{i}=22+27+21+20+25+26+15+23+17+28+21+24+19+28+24 = 330$
## Step3: Calculate the average for Name Brand
Number of data points for Name Brand $n_{name}=17$
Average $\bar{x}_{name}=\frac{S_{name}}{n_{name}}=\frac{439}{17}\approx25.82$
## Step4: Calculate the average for Store Brand
Number of data points for Store Brand $n_{store}=15$
Average $\bar{x}_{store}=\frac{S_{store}}{n_{store}}=\frac{330}{15} = 22$
Since $\bar{x}_{name}>\bar{x}_{store}$, the name brand has more chips on average.

## Step5: Calculate the variance for Name Brand
$$
\begin{align*}
s_{name}^{2}&=\frac{1}{n_{name}- 1}\sum_{i = 1}^{n_{name}}(x_{i}-\bar{x}_{name})^{2}\
\end{align*}
$$
$$
\begin{align*}
\sum_{i = 1}^{17}(x_{i}-\bar{x}_{name})^{2}&=(22 - 25.82)^{2}+(30 - 25.82)^{2}+(25 - 25.82)^{2}+(33 - 25.82)^{2}+(22 - 25.82)^{2}+(27 - 25.82)^{2}+(21 - 25.82)^{2}+(20 - 25.82)^{2}+(25 - 25.82)^{2}+(26 - 25.82)^{2}+(20 - 25.82)^{2}+(23 - 25.82)^{2}+(27 - 25.82)^{2}+(26 - 25.82)^{2}+(29 - 25.82)^{2}+(33 - 25.82)^{2}+(22 - 25.82)^{2}\
&=(-3.82)^{2}+(4.18)^{2}+(-0.82)^{2}+(7.18)^{2}+(-3.82)^{2}+(1.18)^{2}+(-4.82)^{2}+(-5.82)^{2}+(-0.82)^{2}+(0.18)^{2}+(-5.82)^{2}+(-2.82)^{2}+(1.18)^{2}+(0.18)^{2}+(3.18)^{2}+(7.18)^{2}+(-3.82)^{2}\
&=14.5924+17.4724 + 0.6724+51.5524+14.5924+1.3924+23.2324+33.8724+0.6724+0.0324+33.8724+7.9524+1.3924+0.0324+10.1124+51.5524+14.5924\
&=307.7056
\end{align*}
$$
$s_{name}^{2}=\frac{307.7056}{16}\approx19.23$

## Step6: Calculate the variance for Store Brand
$$
\begin{align*}
s_{store}^{2}&=\frac{1}{n_{store}-1}\sum_{i = 1}^{n_{store}}(y_{i}-\bar{x}_{store})^{2}\
\end{align*}
$$
$$
\begin{align*}
\sum_{i = 1}^{15}(y_{i}-\bar{x}_{store})^{2}&=(22-22)^{2}+(27 - 22)^{2}+(21 - 22)^{2}+(20 - 22)^{2}+(25 - 22)^{2}+(26 - 22)^{2}+(15 - 22)^{2}+(23 - 22)^{2}+(17 - 22)^{2}+(28 - 22)^{2}+(21 - 22)^{2}+(24 - 22)^{2}+(19 - 22)^{2}+(28 - 22)^{2}+(24 - 22)^{2}\
&=0^{2}+5^{2}+(-1)^{2}+(-2)^{2}+3^{2}+4^{2}+(-7)^{2}+1^{2}+(-5)^{2}+6^{2}+(-1)^{2}+2^{2}+(-3)^{2}+6^{2}+2^{2}\
&=0 + 25+1+4+9+16+49+1+25+36+1+4+9+36+4\
&=210
\end{align*}
$$
$s_{store}^{2}=\frac{210}{14}=15$

Since $s_{store}^{2}<s_{name}^{2}$, the store brand is more consistent.

# Answer:
(a) C. Yes. The name brand appears to have more chips per cookie.
(c) B. Yes. The store brand has a more consistent number of chips per cookie.

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the sum of chips for Name Brand

Step2: Calculate the sum of chips for Store Brand

Step3: Calculate the average for Name Brand

Step4: Calculate the average for Store Brand

Step5: Calculate the variance for Name Brand

Step6: Calculate the variance for Store Brand

Answer: