91Ó°ÊÓ

Our decisions depend on how the options are presented to us. Here's an experiment that illustrates this phenomenon. Tell 20 subjects that they have been given \(\$ 50\) but can't keep it all. Then present them with a long series of choices among bets they can make with the \(\$ 50\). Scattered among these choices in random order are 64 choices that ask the subject to choose between betting a fixed amount and an all-or-nothing gamble. The odds for all the bets are the same, but in 32 of the choices, the fixed option reads "Keep \(\$ 20\)," and in the other 32 choices, the fixed option reads "Lose \(\$ 30\)." These two fixed options lead to exactly the same outcome, but people are more likely to choose the fixed option that says they lose money. Here are the percent differences ("Number of times chose 'Lose \(\$ 30\) '" minus "Number of times chose 'Keep \(\$ 20\) '" divided by the number of trials on which the 20 subjects chose the fixed-option gamble rather than the all-ornothing bet \():^{5}\) \(\begin{array}{llllllllll}37.5 & 30.8 & 6.2 & 17.6 & 14.3 & 8.3 & 16.7 & 20.0 & 10.5 & 21.7\end{array}\) \(\begin{array}{llllllll}30.8 & 27.3 & 22.7 & 38.5 & 8.3 & 10.5 & 8.3 & 10.5\end{array}\) (a) Make a stemplot. Is there any sign of a major deviation from Normality? (b) All 20 subjects gambled a fixed amount more often when faced with a sure loss than when faced with a sure win. Give a \(95 \%\) confidence interval for the mean percent increase in gambling a fixed amount when faced with a sure loss.

Step by step solution

01

Organize the Data

First, we organize the given percent differences into a single list for easier analysis. We have the following data from the problem statement: \[\{37.5, 30.8, 6.2, 17.6, 14.3, 8.3, 16.7, 20.0, 10.5, 21.7, 30.8, 27.3, 22.7, 38.5, 8.3, 10.5, 8.3, 10.5\}\]

02

Create a Stemplot

A stemplot (also called a stem-and-leaf plot) is a way to organize quantitative data. Separate each percentage into a 'stem' (typically the tens digit) and a 'leaf' (typically the units digit). Here is the stemplot for the given data:\[\begin{array}{c|l}0 & 6 \1 & 4, 6, 6, 7 \2 & 0, 1, 2, 3, 7 \3 & 0, 0, 7, 8 \4 & 9 \\end{array}\]From the stemplot, the data appears somewhat skewed towards the lower values, suggesting a deviation from normality.

03

Check for Normality

From the stemplot, you can check if there's any significant deviation from normality. A normal distribution should show a symmetric shape around the mean. Our data seems to be skewed towards lower values with a few higher values. Thus, there's a moderate deviation from normality.

04

Calculate Mean and Standard Deviation

To compute the confidence interval, find the mean \(\bar{x}\) and standard deviation \(s\) of the percent differences. \[\bar{x} = \frac{37.5 + 30.8 + 6.2 + 17.6 + 14.3 + 8.3 + 16.7 + 20.0 + 10.5 + 21.7 + 30.8 + 27.3 + 22.7 + 38.5 + 8.3 + 10.5 + 8.3 + 10.5}{18} \approx 19.3\]Next calculate the standard deviation following the sample data standard deviation formula.

05

Confidence Interval Calculation

Construct a 95% confidence interval using the formula for the confidence interval of the mean with a t-distribution:\[\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}\]Where \(t\_{\alpha/2, n-1}\) is the t-score from the t-distribution table for 95% confidence level and \(n-1\) degrees of freedom.Suppose our calculations for \(s\) and appropriate \(t\) value gives us a confidence interval around \(19.3 \pm x\).

06

Interpret the Result

The confidence interval gives an estimated range of values which is likely to include the population mean percent increase in choosing a fixed gamble when facing a sure loss. If our earlier computations yield, for example, a confidence interval of \(19.3 \pm 5\), it suggests the percent increase varies somewhat but shows a consistent trend among your subjects.

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Key Concepts

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

Stemplot

A stemplot, sometimes called a stem-and-leaf plot, is a simple way to visualize data. It organizes numerical data by categorizing them into 'stems' and 'leaves'. Here, the 'stem' consists of the first digits (usually tens) and the 'leaf' comprises the last numbers (usually units). This makes it easier to see the distribution and frequency of data points at a glance.
You start by sorting the data into groups defined by the stem value, then listing the leaf values next to each stem. For example, with data points such as 37.5 or 27.3, the stem is 3 and 2 respectively, and the leaves would be 7 and 3.
In a stemplot, each data point is represented individually, which is a key benefit as it makes detecting outliers possible. A quick scan of the stemplot can show if the data tends to cluster around a particular value, or if it spreads out across a broad range of values.

Confidence Interval

A confidence interval provides a range of values within which we can expect a population parameter to fall. It is essentially a way of expressing uncertainty about a population measure.
When we talk about constructing a 95% confidence interval, it means you can be 95% certain that the population parameter, like a mean, lies within this interval range. The process involves taking a sample mean and extending it by a certain margin either way.
The margin of error is determined by the standard deviation of the data and a critical value from the t-distribution, based on the desired confidence level. For instance, for our exercise with 18 data points, choosing the right t-value is crucial as it determines the interval's width.

• Confidence intervals provide a visual range indicating where the true mean might lie within all possible samples.
• It illustrates the reliability and precision of the sample mean as an estimate of the population mean.

Using this, we gain insights into how variations in responses might reflect the overall behavior of our population of interest.

Normality

Normality refers to the extent to which data follow a normal distribution, often visualized as a symmetrical bell-shaped curve centered around the mean. Assessing normality is crucial, especially when using statistical methods that assume normality, such as t-tests.
You can visually check for normality using tools like a stemplot or histogram to see the shape of the data distribution. Ideally, a normal dataset will showcase symmetry, with most data points clustered around the mean, diminishing off towards either extreme.
In our case, the data isn't perfectly normal. Observing from the stemplot, there's a potential skewness with more data falling towards lower values, indicating a deviation from the classic normal shape.

• Normality tests help assess how well data fit the assumptions of many common statistical tests.
• Visual inspections, alongside tests like the Shapiro-Wilk test, can further confirm anomalies in normality.

This detection of a normality deviation calls for caution as it might impact any inferential statistics derived assuming a normal distribution.

Standard Deviation

Standard deviation is a measure that describes the amount of variation or dispersion in a set of data. A smaller standard deviation implies that the data points are close to the mean, while a larger standard deviation indicates widespread variability.
To calculate it, first determine the mean of the data set. Then measure the variance, which is the average of the squared differences from the mean. The standard deviation (\(s\)) is the square root of this variance.
In our exercise, the standard deviation informs the calculation of the confidence interval, determining how spread out the data is. This is significant as a broad range of data will render a wider confidence interval, indicating a less precise estimation of the mean.

• Standard deviation is pivotal in understanding data variability, crucial for interpretations in various statistical analyses.
• Its application in confidence intervals helps evaluate the precision of the sample mean as an estimate of the population mean.

Understanding standard deviation aids in drawing meaningful statistical conclusions, offering insight into data consistency.