Question

Seventy-seven students at the University of Virginia were asked to keep a diary of a conversation with their mothers, recording any lies they told during these conversations (San Luis Obispo Telegram-Tribune, August 16 , 1995 ). It was reported that the mean number of lies per conversation was \(0.5 .\) Suppose that the standard deviation (which was not reported) was \(0.4\). a. Suppose that this group of 77 is a random sample from the population of students at this university. Construct a \(95 \%\) confidence interval for the mean number of lies per conversation for this population. b. The interval in Part (a) does not include 0 . Does this imply that all students lie to their mothers? Explain.

Short answer

Our \(95\% \) confidence interval for the average number of lies told per conversation is calculated around the sample mean of \(0.5\) lies. Although 0 is not included in our confidence interval, this does not mean every student lies to their mothers. This only implies that the average number of lies per conversation for the whole student population is likely to be greater than 0.

Step-by-step solution

Identify Given Values

Identify the sample size \(n = 77\), the sample mean \(\overline{x} = 0.5\), and the sample standard deviation \(s = 0.4\). The confidence level is \(95\% \).

Determine the Confidence Level in Z-Scores

For a \(95\% \) confidence level, the z-value is found using the Standard Normal (Z) Table or commonly used probability values for the normal distribution. The z-value for \(95\% \) confidence level is approximately \(1.96\). This is found by learning that \(95\% \) of the area under the standardized normal curve lies within 1.96 standard deviations of the mean.

Calculate Confidence Interval

Use the formula to calculate the confidence interval. The formula for confidence interval is: \(\overline{x} \pm z \dfrac{s}{\sqrt{n}}\). Substituting in values, the calculation becomes \(0.5 \pm 1.96 \dfrac{0.4}{\sqrt{77}}\). Calculate the upper and lower confidence interval values.

Interpretation of Part a

The result is the \(95\% \) confidence interval for the average number of lies per conversation reported by the students at that University.

Interpretation of Part b

The interval not including 0 means that it is statistically unlikely (less than \(5\% \) chance) that the true average number of lies per conversation is 0. However, this doesn't imply that all students lie to their mothers. It only means that, on average, the students in this sample reported telling some lies. There could be many students who didn't tell any lies at all, so long as there are others who told more than one lie to balance it.

Key concepts

Statistics Education

Understanding statistics is essential for interpreting research findings, making informed decisions based on data, and even becoming a critical consumer of information presented in the media. A reliable statistics education entails learning about various concepts, including data collection, analysis, interpretation, and presentation. When studying confidence intervals, as in our exercise, students learn to estimate the range within which a population parameter is likely to lie with a given level of certainty. This is a key concept in inferential statistics, which allows us to make predictions or inferences about a population based on sample data.

Normal Distribution

A normal distribution, often represented by a bell curve, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In the context of creating a confidence interval, when the sampling distribution of the mean is normally distributed or the sample size is large, the normal distribution can be used to determine how likely it is that the sample mean will fall within a certain range around the actual population mean. This information, combined with the concept of z-scores, lets us calculate the boundaries of the confidence interval for the mean number of lies told by students to their mothers.

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variability or dispersion of a set of values. It reflects how much the elements in the sample differ from the sample mean. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates the values are spread out over a wider range. When computing the confidence interval in our exercise, the sample standard deviation is used to estimate the standard deviation of the population, helping to determine the margin of error and the width of our confidence interval.

z-scores

Z-scores are the number of standard deviations away from the mean a data point is. In the context of confidence intervals, z-scores are used to identify the critical values that correspond to a designated level of confidence. These z-scores, when multiplied by the standard deviation and adjusted for the sample size, provide the margin of error for the interval. For example, a z-score of 1.96 is commonly used for a 95% confidence interval. This tells us that there is a 95% chance that the population mean will fall between 1.96 standard deviations below and above the sample mean.