Q44E A random sample of 50 consumers ... [FREE SOLUTION]

91影视

Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset 91影视 Explanations$ Math

13th Edition 路 888 pages

ISBN: 9780134506593

Chapter 6: Q44E (page 356)

A random sample of 50 consumers taste-tested a new snack food. Their responses were coded (0: do not like; 1: like; 2: indifferent) and recorded as follows:
a. Use an 80% confidence interval to estimate the proportion of consumers who like the snack food.
b. Provide a statistical interpretation for the confidence interval you constructed in part a.

Short Answer

Expert verified

a. The 80% confidence interval is $(0.2170, 0.3830)$

b.One is 80% confident that the true proportion of consumers who like the snack food lies between the interval $(0.2170, 0.3830)$

Step by step solution

Given information

A random sample of 50 consumers taste-tested a new snack food. Their responses were coded according to some criteria (0: do not like; 1: like; 2: indifferent). This has been presented in the table above.

(a) Calculation of 80% confidence interval of estimate the proportion of consumers who like the snack food.

The sample size is n= 50

15 of the data value are a 1, which indicates x=15

$p = \frac{x}{n} = \frac{15}{50} = 0.3$

Here the confidence coefficient is 80%, hence $伪 = 20 %$ .Now from the standard normal distribution table, $z_{\frac{伪}{2}} = 1.28$

The margin of error

$\begin{array}{rcl} E & = & z_{\frac{伪}{2}} 脳 \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \\ = & 1.28 脳 \sqrt{\frac{0.3 脳 (1 - 0.3)}{50}} \\ = & 0.0830 \end{array}$

Hence 80% lower limit: $\hat{p} - E = 0.3 - 0.0830 = 0.2170$

80% upper limit: $\hat{p} + E = 0.3 + 0.0830 = 0.3830$

The 80% confidence interval is $(0.2170, 0.3830)$

(b) Interpretation of confidence interval constructed in part a

One is 80% confident that the true proportion of consumers who like snack food lies between the interval $(0.2170, 0.3830)$

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Short Answer

Step by step solution

Given information

(a) Calculation of 80% confidence interval of estimate the proportion of consumers who like the snack food.

(b) Interpretation of confidence interval constructed in part a

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