Class 3-30-2023: Chapter 7 Part 2 and Review Test
Welcome to a review session focusing on Chapters 6 and 7! This session covers essential concepts and prepares you for the upcoming test. Let's dive in!
Key Concepts Covered
- Sampling Distributions: Understanding how sample statistics vary from sample to sample.
- Central Limit Theorem (CLT): A cornerstone of statistics! The CLT states that for a relatively large sample size, the distribution of the sample mean ($\bar{x}$) is approximately normally distributed, regardless of the distribution of the original variable. The approximation gets better with increasing sample size.
- Standard Error: The standard deviation of the sampling distribution of the sample mean, denoted as $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
- Z-scores: Used to determine the number of standard deviations from the mean.
Review Test Practice Problems
Let's work through some problems similar to what you might see on the test. Remember, practice makes perfect!
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Z-scores and Area: For the standard normal curve, find the z-score(s):
- That have an area 0.250 to its left.
- That has area 0.15 to its right.
- $Z_{0.05}$
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GRE Scores (Normal Distribution): The GRE math section has a mean of 152 and a standard deviation of 8.8. Assuming these scores are normally distributed:
- Obtain and interpret the quartiles.
- Find and interpret the 99th percentile.
- Approximately 68% of students score between what two values? (Hint: Empirical Rule)
- Approximately 95% of students score between what two values? (Hint: Empirical Rule)
- Approximately 99.7% of students score between what two values? (Hint: Empirical Rule)
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Rotavirus Vaccine (Normal Approximation to Binomial): A vaccine is 90% effective. Out of 1500 cases, what's the probability the vaccine will be effective in:
- Exactly 1325 cases?
- At least 1275 cases?
- Between 1050 and 1150 cases, inclusive?
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Pygmy-Possums (Sampling Distribution): Pygmy-possum weights are normally distributed with a mean of 8.3g and a standard deviation of 0.4g.
- Find the standard deviation of the sampling distribution for a sample size of 16.
- Find the percentage of samples of 16 pygmy-possums that have mean weights within 0.125g of the population mean weight of 8.3g.
- Find the percentage of samples of 25 pygmy-possums that have mean weights within 0.125g of the population mean weight of 8.3g.
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Roof Lifespan (Hypothesis Testing): A roof manufacturer claims a 25-year average lifespan with a 3-year standard deviation.
- One roof lasts 23 years. Is this evidence against the claim?
- 100 roofs average 24 years. Is this evidence against the claim?
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Women's Height(Sampling Distribution): In the 1950s the mean height of women in their 20s was 62.6 inches with a standard deviation of 2.88 inches. If the mean height today is the same as that of the 1950s what percentage of all samples of 50 of today's women in their 20s have a mean height of at least 64.24 inches?
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Water Softener Salt(Sampling Distribution): A brand of water-softener salt comes in packages marked \"net weight 40 lb.\" The company that packages the salt claims that the bags contain an average of 40 lb of salt and that the standard deviation of the weights is 1.5 lb.
- Obtain the probability that the weight of one randomly selected bag of water-softener salt will be 41 lb or less, if the company's claim is true.
- Determine the probability that the mean weight of 10 randomly selected bags of water-softener salt will be 41 lb or less, if the company's claim is true.
- If you bought one bag of water-softener salt and it weighed 41 lb, would you consider this evidence that the company's claim is incorrect? Explain your answer.
- If you bought 10 bags of water-softener salt and their mean weight was 41 lb, would you consider this evidence that the company's claim is incorrect? Explain your answer. Assume that the weights are normally distributed.
Remember to review your class notes and practice similar problems. Good luck with your test! You got this!