Chapter 11, Problem 3CA

(a)

To determine

To find: the number of people who were surveyed

Answer to Problem 3CA

There are about 2246 people in the survey.

Explanation of Solution

Given:

Survey Results
Cup	45%
Cone	29%
Sundae	18%
Other	8%
(margin of error $\pm 2.11\%$)

Formula Used:

  $\pm \frac{1}{\sqrt{n}}$

Calculation:

Margin of error = $\pm \frac{1}{\sqrt{n}}$

  $\pm 2.11\%=\pm \frac{1}{\sqrt{n}}\left(a\right)$ Use the formula for margin of error to find the sample size n.

  $\pm 0.0211=\pm \frac{1}{\sqrt{n}}$

Squaring both sides,

  $\begin{array}{l}0.00044521=\frac{1}{n}\\ n=\frac{1}{0.00044521}\\ n\approx \boxed{2246}\end{array}$

Thus, there are about 2246 people in the survey.

Conclusion:

There are about 2246 people in the survey.

(b)

To determine

To find: reason why the given conclusion is inaccurate to draw from the given data

Answer to Problem 3CA

More than one-fourth of the adults who were surveyed prefer to have ice-cream in cone.

Explanation of Solution

Given:

“Adults do not prefer to eat their ice cream in cone”

Calculation:

It is inaccurate to draw the conclusion that adults do not prefer to eat their ice cream in cone because from the given data it is clear that more than one-fourth of the adults who were surveyed prefer to have ice-cream in cone.

Conclusion:

More than one-fourth of the adults who were surveyed prefer to have ice-cream in cone.

(c)

To determine

To find: the probability that at least three prefer to eat ice cream in a cone

Answer to Problem 3CA

The probability that at least three prefer to eat ice cream in a cone is $0.24$

Explanation of Solution

Calculation:

Now, here probability of success is 0.29, since 29% people in survey prefers ice-cream in cone.

Now, to find probability that at least 3 out of 6 people prefer to have ice cream in cone is given by

This is a binomial experiment and for a binomial experiment, the probability of exactly k successes in n trials is given by

  $P\left(ksuccesses\right){=}_n{C}_k{p}^2{\left(1-p\right)}^{n-k}$

Here $p=0.29,n=6andk=3$ and need to find $P\left(x\ge 3\right)$ .

Now,

  $\begin{array}{l}P\left(x\ge 3\right)=P\left(x=3\right)+P\left(x=4\right)+P\left(x=5\right)+P\left(x=6\right)\hfill \\ =0.24\hfill \end{array}$

Conclusion:

The probability that at least three prefer to eat ice cream in a cone is $0.24$

(d)

To determine

To find: the reason why the given conclusion is incorrect

Answer to Problem 3CA

This conclusion might be incorrect as a sample of size 6 is very small and it will have a large margin of error.

Explanation of Solution

Given:

Four of the six people eat ice cream in cone

Calculation:

This conclusion might be incorrect as a sample of size 6 is very small and it will have a large margin of error.

Conclusion:

This conclusion might be incorrect as a sample of size 6 is very small and it will have a large margin of error.

(e)

To determine

To find: the margin of error for the survey

Answer to Problem 3CA

Margin of error is about $\pm 40.8\%$

Explanation of Solution

Calculation:

Use the formula for margin of error

Margin of error =

  $\begin{array}{l}\pm \frac{1}{\sqrt{n}}\\ =\pm \frac{1}{\sqrt{6}}\\ \approx 0.408\end{array}$

Thus, margin of error is about $\pm 40.8\%$

Conclusion:

Margin of error is about $\pm 40.8\%$