A random sample of 50 consumers taste-tested a new snack...

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: $$ \begin{array}{llllllllll} \hline 1 & 0 & 0 & 1 & 2 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 2 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 \\ \hline \end{array} $$ a. Use an $80 \%$ confidence interval to estimate the proportion of consumers who like the saack food. b. Provide a statistical interpretation for the confidence interval you constructed in part a.

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:
$$
\begin{array}{llllllllll}
\hline 1 & 0 & 0 & 1 & 2 & 0 & 1 & 1 & 0 & 0 \\
0 & 1 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\
0 & 2 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 \\
\hline
\end{array}
$$
a. Use an $80 \%$ confidence interval to estimate the proportion of consumers who like the saack food.
b. Provide a statistical interpretation for the confidence interval you constructed in part a.

Key Concepts

Interpretation of the Confidence Interval

The interpretation of a confidence interval is that, if one were to take many samples and construct a new interval from each sample in the same way, about 80% of those intervals would contain the true population proportion. It is important to note that this statement applies to the long-run behavior of the interval construction process rather than the probability that a single computed interval contains the parameter.

Critical Value and Confidence Level

The critical value is a factor used in the calculation of the margin of error, determined by the desired confidence level (80% in this case). It scales the standard error to reflect the width of the confidence interval needed to capture the true parameter with the stated degree of confidence.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of a population parameter with a specified level of confidence. It provides an estimated interval of uncertainty around a sample statistic and indicates how precise that estimate might be.

Proportion Estimate

The proportion estimate is the statistic calculated from sample data that represents the fraction of the sample exhibiting a particular characteristic. It serves as the point estimate for the true population proportion, which is the parameter of interest.

Margin of Error

The margin of error is the amount added and subtracted from the sample estimate to create the confidence interval. It accounts for sampling variability and is computed by multiplying a critical value (obtained from the relevant distribution) by the standard error of the estimate.

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