Zenon is investigating whether students at his school prefer name-brand …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

Zenon is investigating whether students at his school prefer name-brand potato chips over generic ones. He randomly selected 50 students, having each try both types of chips in random order. Of these students, 32 preferred name-brand chips. Zenon wants to conduct a test at the α = 0.05 significance level, testing H0: p = 0.5 versus Ha: p > 0.5 (where p is the proportion of all students at his school who prefer name-brand chips). a. Are the conditions for performing the test satisfied? b. What is the standardized test statistic?

Short Answer

To conduct the test, ensure conditions like a random sample of 50 students, independence of preferences, and that the sample constitutes less than 10% of the population are met. Calculate the sample proportion of students preferring name-brand chips (0.64), and use it to derive the standardized test statistic ((Z approx 1.981)) for hypothesis testing.

Step-by-Step Solution

Step 1: Verify Conditions for the Test

Before conducting the test, ensure that the necessary conditions are satisfied. This involves confirming:

  • Random Sample: Zenon must have randomly selected 50 students from the entire population to meet this criterion.
  • Independence: Each student’s preference must be independent of others, which is satisfied since each student tries the chips in random order.
  • Sample Size: The sample size should be less than 10% of the total population, ensuring that 50 students constitute less than 10% of the school’s population.

Step 2: Calculate the Sample Proportion

To determine the sample proportion, calculate the number of students who preferred the name-brand chips. In Zenon’s case, this is done by:

  • Finding the proportion: Divide the number of students who prefer the name-brand chips (32) by the total number of sampled students (50).
  • Formula Application: Thus, the sample proportion is calculated as 0.64 (i.e., 32/50).
  • Hypothesized Value: Set the hypothesized proportion under the null hypothesis, which is 0.5.

Step 3: Calculate the Standardized Test Statistic

The standardized test statistic is vital for making inferences about the population proportion. This involves:

  • Applying the Formula: Use the formula (Z = frac{hat{p} – p_0}{sqrt{frac{p_0(1 – p_0)}{n}}}) with (hat{p} = 0.64), (p_0 = 0.5), and (n = 50).
  • Substituting Values: After substituting the values into the formula, calculate the Z value systematically.
  • Result: The computation yields (Z approx 1.981), which will be compared against critical values for hypothesis testing.

Related Concepts

Random Sample

A sample that is selected in such a way that every member of the population has an equal chance of being chosen, ensuring that the sample is representative of the population as a whole.

Independence

A condition in which the selection or preference of one sample unit does not influence or affect the selection or preference of another unit, allowing each data point to be treated independently.

Sample Proportion

The ratio of the number of success observations (e.g., preferences for a specific option) in a sample to the total number of observations in that sample, used to estimate the corresponding proportion in the entire population.

Table Of Contents
  1. Zenon is investigating whether students at his school prefer name-brand potato chips over generic ones. He randomly selected 50 students, having each try both types of chips in random order. Of these students, 32 preferred name-brand chips. Zenon wants to conduct a test at the α = 0.05 significance level, testing H0: p = 0.5 versus Ha: p > 0.5 (where p is the proportion of all students at his school who prefer name-brand chips). a. Are the conditions for performing the test satisfied? b. What is the standardized test statistic?

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