Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

문제 6.1.41

6장, 문제 6.1.41

In Exercise 37, does it seem likely that the population mean could be greater than \$70? Explain.

검증된 단계별 안내

Step 1: Identify the context of the problem. The question is asking whether the population mean (μ) could be greater than \$70. This typically involves hypothesis testing or confidence interval analysis.

Step 2: Determine the statistical method to use. If the problem provides sample data (e.g., sample mean, standard deviation, and sample size), you can construct a confidence interval for the population mean or perform a hypothesis test.

Step 3: If constructing a confidence interval, calculate the margin of error using the formula:

E = z

_α/2 ⋅ (σ/n), where

z

_α/2 is the critical value,

σ

is the population standard deviation (or sample standard deviation if

σ

is unknown), and

n

is the sample size.

Step 4: If performing a hypothesis test, set up the null hypothesis

H_{0} : μ ≤ 70

and the alternative hypothesis

H_{a} : μ > 70

. Use the test statistic formula:

z = (x̄ - μ ₀) / (σ / \sqrt{n})

, where

x̄

is the sample mean,

μ ₀

is the hypothesized mean, and

σ

is the standard deviation.

Step 5: Interpret the results. For a confidence interval, check if \$70 is within the interval. If it is not, it suggests the population mean could be greater than $70. For a hypothesis test, compare the p-value to the significance level (e.g., 0.05). If the p-value is small, reject the null hypothesis, indicating that the population mean is likely greater than $70.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Population Mean

The population mean is the average of all values in a population, representing a central point around which data points are distributed. It is a key parameter in statistics, often denoted by the symbol μ. Understanding the population mean is crucial for making inferences about the entire population based on sample data.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample to support a specific claim about a population parameter. In this context, one might test the hypothesis that the population mean is greater than $70. This involves comparing sample data against a null hypothesis to assess the likelihood of observing the data if the null hypothesis is true.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, if a confidence interval for the population mean does not include $70, it suggests that the mean is unlikely to be greater than this value. Understanding confidence intervals helps in assessing the precision and reliability of estimates.