A random sample of 120 shoppers was asked to compare two new...

Name: Problem 28, Chapter 14: Statistics for Business and Economics
Uploaded: 2022-09-28T09:11:58-08:00
Duration: 1 min 43 s
Channel: Adriano Chikande
Description: A random sample of 120 shoppers was asked to compare two new energy drinks. Sixty-five sample members preferred energy drink A, 53 preferred energy drink B, and 2 expressed no preference. Use the normal approximation to determine if there is an overall preference for either energy drink.

A random sample of 120 shoppers was asked to compare two new energy drinks. Sixty-five sample members preferred energy drink A, 53 preferred energy drink B, and 2 expressed no preference. Use the normal approximation to determine if there is an overall preference for either energy drink.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population parameter based on sample data. It involves formulating a null hypothesis (usually representing no effect or no difference) and an alternative hypothesis (indicating the presence of an effect or difference) to determine if there is enough evidence to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis is the default assumption that no significant effect or difference exists between groups—in this case, that there is no preference between the two energy drinks. The alternative hypothesis, conversely, posits that a significant preference exists, meaning one energy drink is favored over the other.

Normal Approximation to the Binomial Distribution

Normal approximation to the binomial distribution is a method used when sample sizes are large enough such that the binomial distribution of a discrete event can be approximated by the continuous normal distribution. This simplifies calculations in hypothesis testing for proportions by allowing the use of z-scores.

Z-Test for Proportions

A z-test for proportions is used to determine if there is a statistically significant difference between observed proportions and expected proportions under the null hypothesis. This test calculates a z-test statistic by comparing the sample proportion to the hypothesized proportion and dividing the difference by the standard error derived from a normal approximation.

Sampling Distribution of Proportions

The sampling distribution of proportions is the probability distribution of sample proportions over numerous samples from the same population. It is central to applying the normal approximation in hypothesis testing, as it allows for the estimation of variability (standard error) in sample proportions, facilitating the calculation of test statistics like the z-score.

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