Math Problem Statement
Solution
It looks like you've uploaded an image showing a problem related to hypothesis testing, specifically a Z-test for a population proportion.
Here's a breakdown of the problem:
- Claim: 35% of students will find a match the first time using a school-sponsored roommate-matching website.
- Sample size (n): 170 students.
- Sample result (x): 51 students found a match on their first time.
- Hypothesis test: To test if the actual proportion is different from the claimed 35%.
You are asked to compute the Z-test statistic for this hypothesis test.
Formula for Z-test statistic:
Where:
- is the sample proportion.
- is the hypothesized population proportion.
- is the sample size.
- is the number of successes (students who found a match).
Let's calculate this step by step.
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Compute the sample proportion:
-
Use the hypothesized proportion and the sample size to compute the standard error:
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Calculate the Z-statistic.
I'll do the math now.The calculated Z-test statistic is approximately .
This value suggests how many standard deviations the sample proportion is away from the hypothesized proportion of 35%.
Would you like further details on interpreting this result or have any additional questions?
Here are 5 related questions you could explore:
- How do we interpret the Z-test statistic in terms of rejecting or failing to reject the null hypothesis?
- What is the significance level typically used in hypothesis testing, and how does it affect the conclusion?
- How would the result change if the sample size were larger or smaller?
- Can this Z-test be applied to small samples, and if not, what test would be more appropriate?
- How does the choice of a one-tailed vs. two-tailed test affect the result?
Tip: Remember to compare the Z-statistic with the critical value from the Z-table to make a decision about the null hypothesis.
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Math Problem Analysis
Mathematical Concepts
Statistics
Hypothesis Testing
Proportions
Formulas
Z = (p̂ - p₀) / sqrt((p₀ * (1 - p₀)) / n)
p̂ = x / n
Theorems
Central Limit Theorem
Suitable Grade Level
Grades 11-12
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