The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)

Construct a 99% confidence interval for the population standard deviation.

검증된 단계별 안내

Step 1: Recognize that the problem involves constructing a confidence interval for the population standard deviation. Since the population is normally distributed, we will use the chi-square distribution for this calculation.

Step 2: Identify the given values: the sample size (n = 26), the sample standard deviation (s = \$31), and the confidence level (99%).

Step 3: Determine the degrees of freedom (df) for the chi-square distribution. The formula is:

df = n - 1

. Substitute the sample size to calculate df.

Step 4: Find the critical chi-square values for the 99% confidence level. Use a chi-square table or statistical software to find the values for

χ²_{α/2}

(upper critical value) and

χ²_{1-α/2}

(lower critical value), where

α = 1 - 0.99 = 0.01

Step 5: Use the formula for the confidence interval of the population standard deviation:

\(\sqrt{\frac{(n-1)s^2}{χ²_{α/2}\)}} \(\leq\) \(\sigma\) \(\leq\) \(\sqrt{\frac{(n-1)s^2}{χ²_{1-α/2}\)}}

. Substitute the values for n, s, and the critical chi-square values to calculate the interval.

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영상 길이:

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주요 개념

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

Confidence Interval

A confidence interval is a range of values, derived from a sample statistic, that is likely to contain the population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population parameter.

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion in a set of sample data points. It quantifies how much the individual data points deviate from the sample mean. In this context, it is used to estimate the variability of room rates among the sampled hotels, which is crucial for constructing the confidence interval.

Chi-Square Distribution

The Chi-Square distribution is a statistical distribution that is used to estimate the variance of a population based on sample data. When constructing confidence intervals for population variances or standard deviations, the Chi-Square distribution is applied, particularly when the population is normally distributed, as is the case in this question.

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)Construct a 99% confidence interval for the population standard deviation.