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The paper "The Effects of Adolescent Volunteer Activities on the Perception of Local Society and Community Spirit Mediated by Self-Conception" (Advanced Science and Technology Letters [2016]: 19-23) describes a survey of a large representative sample of middle school children in South Korea. One question in the survey asked how much time per year the children spent in volunteer activities. The sample mean was 14.76 hours and the sample standard deviation was 16.54 hours. a. Based on the reported sample mean and sample standard deviation, explain why it is not reasonable to think that the distribution of volunteer times for the population of South Korean middle school students is approximately normal. b. The sample size was not given in the paper, but the sample size was described as "large." Suppose that the sample size was 500 . Explain why it is reasonable to use a one-sample \(t\) confidence interval to estimate the population mean even though the population distribution is not approximately normal. c. Calculate and interpret a confidence interval for the mean number of hours spent in volunteer activities per year for South Korean middle school children. (Hint: See Example 12.7.)

Step by step solution

01

Analyze the Distribution of Volunteer Times

Based on the reported sample mean of 14.76 hours and sample standard deviation of 16.54 hours, it is not reasonable to think that the distribution of volunteer times for the population is approximately normal. This is because the standard deviation is larger than the mean, which suggests that the distribution is likely to be highly skewed and not symmetric like a normal distribution.

02

Explain the Reason for Using a One-Sample t Confidence Interval

Even though the population distribution is not approximately normal, it is reasonable to use a one-sample t confidence interval because the sample size is large (i.e., 500). The Central Limit Theorem states that, for large sample sizes, the sampling distribution of sample means will be approximately normal even if the population distribution is not normal. Therefore, we can use a t-test to estimate the population mean.

03

Calculate the Confidence Interval

To calculate the confidence interval, we first need to find the standard error of the mean (SEM). The SEM is given by \(\frac{s}{\sqrt{n}}\), where 's' is the sample standard deviation and 'n' is the sample size. In this case, s = 16.54 and n = 500. SEM = \(\frac{16.54}{\sqrt{500}}\) = 0.7403 Now we can find the critical t-value. For a 95% confidence interval with 499 degrees of freedom (500 - 1), we can use a t-table or a calculator. The critical t-value for this confidence interval is approximately 1.96. Next, we'll find the margin of error by multiplying the critical t-value by the standard error of the mean: Margin of Error = 1.96 * 0.7403 = 1.4508 Now we can calculate the confidence interval as: Lower limit = Sample Mean - Margin of Error = 14.76 - 1.4508 = 13.3092 Upper limit = Sample Mean + Margin of Error = 14.76 + 1.4508 = 16.2108

04

Interpret the Confidence Interval

We can interpret the confidence interval as follows: We are 95% confident that the mean number of hours spent in volunteer activities per year for South Korean middle school children lies between 13.31 hours and 16.21 hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

In statistics, a confidence interval (CI) provides a range of values that is likely to contain a population parameter with a certain level of confidence. For instance, in the case of the South Korean middle school students' volunteer hours, a 95% confidence interval tells us that we are 95% sure that the true average number of hours falls within the calculated range.

• The formula to calculate the confidence interval for a mean is \( ext{Sample Mean} \pm \text{Margin of Error}\).
• The Margin of Error is calculated by multiplying the critical value from the t-distribution by the standard error of the mean (SEM).

For example, if the sample mean is 14.76 hours and the margin of error is 1.4508 hours, the confidence interval is between 13.3092 and 16.2108 hours. This interval gives researchers insight into the population without needing to survey every individual, streamlining the analysis process.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental concept in statistics that allows us to make inferences about a population based on a sample, even if the population distribution is not normal. This theorem states that the sampling distribution of the sample mean will be approximately normally distributed, provided the sample size is sufficiently large.

• The larger the sample size, the more reliable the approximation of the normal distribution.
• This theorem enables the use of the t-distribution to calculate confidence intervals and conduct hypothesis tests.

In the exercise, even though the population distribution of volunteer hours is skewed, the large sample size of 500 helps ensure that the sample mean follows an approximately normal distribution. This allows us to apply inferential statistical methods such as confidence intervals.

Sample Mean

The sample mean is the average of all observations in a sample and is represented by the symbol \(\bar{x}\). It serves as an estimate of the population mean, providing a central measure of the data collected from a sample.

• It is calculated by summing all sample observations and dividing by the total number of observations \(n\).
• For the middle school students' volunteer hours, the sample mean is 14.76 hours.

The sample mean aggregates the volunteer hours reported by students, allowing researchers to extrapolate insights into the broader population. This statistic forms the foundation for further analysis, including calculating confidence intervals and testing hypotheses.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much individual data points in a distribution deviate from the mean. In the context of the exercise, the sample standard deviation is given as 16.54 hours.

• A high standard deviation suggests that data points are spread out over a larger range of values.
• The formula for standard deviation is \(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\), where \(x_i\) represents each data point.

In the case of the students' volunteer activity, the standard deviation being greater than the mean indicates a skewed distribution, not typical of a standard normal distribution. Nonetheless, it is instrumental in calculations such as the standard error, which is pivotal for constructing confidence intervals.