Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

문제 6.1.39

6장, 문제 6.1.39

In Exercise 35, would it be unusual for the population mean to be over \$1500? Explain.

검증된 단계별 안내

Step 1: Identify the key components of the problem. The question is asking whether it would be unusual for the population mean to exceed \$1500. To determine this, we need to assess the probability of the population mean being greater than \$1500 using the concept of z-scores and the standard normal distribution.

Step 2: Recall the formula for the z-score for a population mean:

z = (μ - μ

₀) / (σ / n), where

μ

is the population mean,

μ

₀ is the hypothesized mean (in this case, \$1500),

σ

is the population standard deviation, and

n

is the sample size.

Step 3: Substitute the given values into the z-score formula. If the problem does not provide the population standard deviation (

σ

) or sample size (

n

), you will need to refer to Exercise 35 for these details.

Step 4: Use the z-score obtained to find the corresponding probability from the standard normal distribution table. This will give you the likelihood of the population mean being less than or equal to \$1500. To find the probability of the mean being greater than \$1500, subtract this value from 1.

Step 5: Interpret the result. If the probability of the population mean being greater than \$1500 is very small (commonly less than 0.05), it would be considered unusual for the population mean to exceed \$1500. Otherwise, it would not be unusual.

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주요 개념

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Population Mean

The population mean is the average of all values in a given population. It is a measure of central tendency that provides insight into the overall characteristics of the population. Understanding the population mean is crucial for determining how individual data points relate to the average, which can help assess whether a specific value, like $1500, is typical or unusual.

Statistical Significance

Statistical significance refers to the likelihood that a result or relationship observed in data is not due to random chance. In the context of the population mean, determining whether a mean over $1500 is unusual involves assessing how far this value is from the expected mean and whether this deviation is statistically significant, often using hypothesis testing.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. Many statistical analyses assume normality, and understanding this concept helps in evaluating whether a population mean over $1500 falls within a typical range or is an outlier, based on the distribution of the data.

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Which statistic is the best unbiased estimator for μ?

a. s

b. xbar

c. the median

d. the mode

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In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.

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