Math Problem Statement

According to the Census​ Bureau, 18.1​% of the people in the United States are of Hispanic or Latino origin. One county supervisor believes her county has a different proportion of Hispanic people than the nation as a whole. She looks at their most recent survey​ data, which was a random sample of 421 county​ residents, and found that 53 of those surveyed are of Hispanic origin. Complete parts​ (a) through​ (d) below. Question content area bottom Part 1 ​a) State the hypotheses. Let the proportion of people in the county that are of Hispanic origin be denoted by p. Choose the correct answer below. A.Upper H 0 : p greater than or equals 0.181 Upper H Subscript Upper A Baseline : p less than 0.181 Upper H 0 : p greater than or equals 0.181 Upper H Subscript Upper A Baseline : p less than 0.181 B.Upper H 0 : p less than 0.181 Upper H Subscript Upper A Baseline : p equals 0.181 Upper H 0 : p less than 0.181 Upper H Subscript Upper A Baseline : p equals 0.181 C.Upper H 0 : p equals 0.181 Upper H Subscript Upper A Baseline : p not equals 0.181 Upper H 0 : p equals 0.181 Upper H Subscript Upper A Baseline : p not equals 0.181 Your answer is correct.D.Upper H 0 : p equals 0.181 Upper H Subscript Upper A Baseline : p less than 0.181 Upper H 0 : p equals 0.181 Upper H Subscript Upper A Baseline : p less than 0.181 Part 2 ​b) Name the model and check appropriate conditions for a hypothesis test. What kind of test is​ this? Difference in differences test ​Two-proportion t test ​One-proportion z test Your answer is correct. Proportional t test Part 3 Check the appropriate conditions for a hypothesis test. Select the correct choice below and fill in the answer boxes within your choice. ​(Round to one decimal place as​ needed.) A. The 421 residents were a random sample from the county of interest. 421 is likely less than​ 10% of the population of a county. One expects np 0equals    enter your response here successes and nq 0equals    enter your response here ​failures, which are both more than 10. B. The 421 residents were a random sample from the county of interest. The sample size is more than 100. One expects np 0equals    enter your response here successes and nq 0equals    enter your response here ​failures, which are both more than 10. C. The 421 residents were a random sample from the county of interest. The standard deviation of p is less than 0.5. One expects np 0equals    enter your response here successes and nq 0equals    enter your response here ​failures, which are both more than 20.

Solution

Let's go step-by-step through the parts of the question.

Part 1: State the Hypotheses

Since the county supervisor wants to determine whether the proportion of Hispanic or Latino residents in her county is different from the national proportion (0.181), we are conducting a two-tailed test.

  • Null Hypothesis H0H_0: The proportion of Hispanic people in the county is equal to the national proportion.

    • H0:p=0.181H_0: p = 0.181

  • Alternative Hypothesis HAH_A: The proportion of Hispanic people in the county is not equal to the national proportion.

    • HA:p≠0.181H_A: p \neq 0.181

Correct Answer:
C. H0:p=0.181H_0: p = 0.181 and HA:p≠0.181H_A: p \neq 0.181

Part 2: Name the Model and Identify the Test

Since we are testing a single proportion from the population (comparing the sample proportion to the national proportion), the appropriate test is a one-proportion z-test.

Correct Answer:
One-proportion z-test

Part 3: Check Conditions for Hypothesis Test

For a one-proportion z-test, the conditions are:

  1. Random Sample: The 421 residents were selected randomly.
  2. Independence Condition: 421 is likely less than 10% of the county’s population.
  3. Success-Failure Condition: Both the expected number of successes np0n p_0 and failures n(1−p0)n (1 - p_0) must be at least 10.

Now, let’s calculate the expected number of successes and failures based on the null hypothesis p0=0.181p_0 = 0.181.

np0=421×0.181=76.2n p_0 = 421 \times 0.181 = 76.2 n(1−p0)=421×(1−0.181)=421×0.819=344.8n (1 - p_0) = 421 \times (1 - 0.181) = 421 \times 0.819 = 344.8

Both values are greater than 10, so the success-failure condition is satisfied.

Correct Answer:
A. The 421 residents were a random sample from the county of interest. 421 is likely less than 10% of the population of a county. One expects np0=76.2n p_0 = 76.2 successes and nq0=344.8n q_0 = 344.8 failures, which are both more than 10.

Would you like to proceed with calculations for the z-test or have any other questions?

5 Follow-up Questions:

  1. How do you calculate the z-statistic for a one-proportion z-test?
  2. What is the critical value for a two-tailed z-test at a 5% significance level?
  3. How do you interpret p-values in hypothesis testing?
  4. What assumptions underlie the use of a z-test versus a t-test?
  5. How would the hypothesis change for a one-tailed test?

Tip: In hypothesis testing, always compare the p-value with the significance level (α\alpha) to determine whether to reject the null hypothesis.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Proportions
One-Proportion Z-Test

Formulas

Z-test formula for proportion: z = (p̂ - p0) / √(p0 * (1 - p0) / n)

Theorems

Central Limit Theorem

Suitable Grade Level

College Level (Statistics Course)

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