91Ó°ÊÓ

In June 2009 , Harris Interactive conducted its Great Schools Survey. In this survey, the sample consisted of 1086 adults who were parents of school-aged children. The sample was selected in a way that makes it reasonable to regard it as representative of the population of parents of school-aged children. One question on the survey asked respondents how much time (in hours) per month they spent volunteering at their children's school during the previous school year. The following summary statistics for time volunteered per month were given: \(n=1086 \quad \bar{x}=5.6 \quad\) median \(=1\) a. What does the fact that the mean is so much larger than the median tell you about the distribution of time spent volunteering per month? b. Based on your answer to Part (a), explain why it is not reasonable to assume that the population distribution of time spent volunteering is approximately normal. c. Explain why it is appropriate to use the \(t\) confidence interval to estimate the mean time spent volunteering for the population of parents of school-aged children even though the population distribution is not approximately normal. d. Suppose that the sample standard deviation was \(s=\) 5.2. Compute and interpret a \(98 \%\) confidence interval for \(\mu,\) the mean time spent volunteering for the population of parents of school-aged children.

Step by step solution

01

Interpreting the Mean and Median

In a symmetric distribution, the mean and the median are almost equal. However, in this case, the mean of \(5.6\) is much larger than the median of \(1\). This suggests that the distribution may be skewed to the right, also known as positively skewed distribution.

02

Reasoning the Shape of the Distribution

Given the mean is larger than the median, it is safe to infer that there are a few very large values in the data set which have pulled the mean up. Such a distribution would have a longer tail to the right, indicating a positive skew. And because of this skew, it is not reasonable to assume the population distribution of time spent volunteering is approximately normal, which is symmetric.

03

Justifying the Use of t Confidence Interval

Although the distribution is not normal, the \(t\) confidence interval can still be used to estimate the mean time due to Central Limit Theorem. The Central Limit Theorem tells us that for large sample sizes (\(n > 30\)), the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. Here, the sample size is quite large (\(n = 1086\)), hence the use of \(t\) confidence interval is justified.

04

Calculating the Confidence Interval

For \(98\%\) confidence interval, the value of \(t\) for \(1085\) degrees of freedom (since \(df = n-1\)) is approximately \(2.614\). The confidence interval for the mean \(\mu\) can then be calculated using the formula \(\bar{x} \pm t(s/\sqrt{n})\), where \(\bar{x} = 5.6\), \(s = 5.2\) and \(n = 1086\). Plugging the values we get: \(5.6 \pm 2.614*(5.2/\sqrt{1086})\).

05

Interpret the Confidence Interval

This confidence interval gives us a range of values for the mean time spent volunteering by parents of school-aged children. We can be \(98\%\) sure that the true mean falls within this interval.

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Key Concepts

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

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in the field of statistics that explains why many distributions tend to approximate a normal distribution as the sample size becomes large, regardless of the shape of the original population distribution. This theorem is particularly useful when analyzing sample means.

According to the CLT, when you take a large number of independent, random samples from a population and calculate their means, those means will form a distribution that is normally distributed or bell-shaped. This holds true even if the population distribution is skewed, uniform, or has another shape entirely. There's one crucial condition: the sample size should be sufficiently large, usually considered as larger than 30 samples.

Thus, when statisticians are faced with a skewed distribution or an unknown population distribution, they can still make inferences about the population mean using the CLT. For example, even if the actual distribution of time parents spent volunteering at school is not normal, as long as the sample size is large (as with our 1086 parents), the sampling distribution of the mean time spent volunteering will be approximately normal. This is why it's possible to use parametric tests, like the t-test or constructing a t confidence interval to estimate the population mean.

t confidence interval

A t confidence interval is a type of interval estimate used to infer the true mean of a population based on a sample mean. The 't' refers to the t-distribution, which is a type of probability distribution that is symmetrical and bell-shaped, like the normal distribution, but has heavier tails. These heavier tails take into account the added uncertainty that arises from estimating the population standard deviation with the sample standard deviation, particularly in smaller samples.

The formula for computing a t confidence interval when the population standard deviation is not known is: \[\bar{x} \pm t\left(\frac{s}{\sqrt{n}}\right)\],where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the sample size, and \(t\) is the t-score from the t-distribution for a specified confidence level and degrees of freedom (df), which is typically \(n-1\).

In practice, you would utilize a t-table or statistical software to determine the appropriate t-value for your confidence level and sample size. This is essential because using a t-distribution accounts for both the uncertainty of the sample data and the small sample size, thus providing a more accurate interval estimate for the population mean.

Skewed Distribution

A skewed distribution is one in which the values of the dataset are not evenly distributed around the mean. In other words, one tail of the distribution is longer than the other. If the tail on the right side is longer, the distribution is said to be 'right-skewed' or 'positively skewed.' Conversely, if the tail on the left side is longer, it is 'left-skewed' or 'negatively skewed.'

In the context of the Great Schools Survey example, a mean much larger than the median suggests a positive skew. This often occurs in data where there are a few very high values that drag the mean upwards, creating a long tail to the right. In such situations, the median, which is less sensitive to outliers, is often considered a better measure of central tendency.

Understanding the type of skewness is crucial when performing data analysis as it impacts the choice of statistical methods to use. Some statistical techniques, like those assuming normality, might not be appropriate for skewed distributions. But as mentioned earlier, thanks to the Central Limit Theorem, with a large sample size, the distribution of the sample means will be approximately normally distributed, allowing us to use methods like the t confidence interval to estimate population parameters.