Math Problem Statement
Solution
The question in the image asks you to select the appropriate test statistic for a hypothesis test based on different sampling scenarios, specifically focusing on testing the population mean. Let's break down the steps for each part of the question:
General Problem Setup:
- The hypothesis test concerns whether the population mean daily profit of a convenience store is less than 495 dollars.
- A sample of daily profits is drawn, with a mean of 503 dollars and a standard deviation of 76 dollars.
Part (a) - Sample size 14, normally distributed population with unknown standard deviation:
When the population standard deviation is unknown, and the sample size is small (n < 30), the t-test should be used. Therefore:
- Test statistic: t
- Formula:
Where:
- (sample mean)
- (hypothesized population mean)
- (sample standard deviation)
- (sample size)
Part (b) - Sample size 110, non-normally distributed population with known standard deviation of 75:
When the sample size is large (n > 30), even if the population is non-normal, by the Central Limit Theorem, the sampling distribution of the mean is approximately normal. Since the population standard deviation is known, we use the z-test:
- Test statistic: z
- Formula:
Where:
- (sample mean)
- (hypothesized population mean)
- (population standard deviation)
- (sample size)
Would you like me to calculate the actual test statistics for both parts?
Here are 5 related questions to help expand on this:
- How do we determine whether to use the z-test or t-test in hypothesis testing?
- Why is the Central Limit Theorem important in large sample hypothesis testing?
- What assumptions do we make when using the t-test?
- How do we interpret the p-value in hypothesis testing?
- What is the difference between a one-tailed and two-tailed test?
Tip: The t-distribution is more spread out than the z-distribution when sample sizes are small, which accounts for the increased uncertainty in estimating the population standard deviation.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
T-test
Z-test
Central Limit Theorem
Normal Distribution
Formulas
t = (x̄ - μ) / (s / √n)
z = (x̄ - μ) / (σ / √n)
Theorems
Central Limit Theorem
Suitable Grade Level
College Level (Statistics)
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