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Chapter 2: Problem 101

Find and interpret the z-score for the data value given. The value 8.1 in a dataset with mean 5 and standard deviation 2

Short Answer

Expert verified
The z-score is 1.55, which means the data value 8.1 is 1.55 standard deviations above the mean of the dataset.

Step by step solution

01

Identify the given values in the problem

From the problem, the following values are given: Value = 8.1, Mean = 5, Standard deviation = 2
02

Apply the values to the Z-score formula

The Z-score formula is \(z = \frac{Value - Mean}{Standard deviation}\). Now, we substitute the given values into the formula. \(z = \frac{8.1 - 5}{2}\)
03

Calculate the Z-score

Performing the calculation in the numerator first, then dividing, we get: \(z = \frac{3.1}{2} = 1.55\)
04

Interpret the Z-score

A Z-score of 1.55 means the data value 8.1 is 1.55 standard deviations above the mean of the dataset

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Key Concepts

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

Standard Deviation
Understanding the standard deviation is crucial when interpreting data. It is a measure that tells us how much variation or dispersion there is from the average (mean) of a set of data.

Think of it like this: if we have a group of people and we measure their heights, the average height might tell us one part of the story, but knowing how much the heights vary tells us a lot more. If everyone is around the same height, we have a small standard deviation. If we have some really tall and some really short people, the standard deviation will be larger, signaling a wide range of differences. In the context of the exercise, a standard deviation of 2 means that the values in the dataset typically deviate from the mean by a distance of 2 units.
Data Value
The term 'data value' refers to a single piece of information that's been recorded. In our daily lives, we compare these values to make decisions. For instance, if you're looking at temperatures over a week to decide what to wear, each day's temperature is a data value.

In the exercise, the data value in question is 8.1. This is a specific observation or measurement from the dataset we are analyzing. It's like noticing on one particular day that the temperature spiked to 8.1 degrees above what's usual for this time of year, and you're trying to figure out how significant that spike is.
Mean
The mean, often referred to as the average, is a central value of a set of numbers. It's calculated by adding up all the values and then dividing by the number of values. In our previous analogy, it's like finding out the average temperature for the week.

The mean gives us a reference point to compare individual data values. In the exercise, the mean of the dataset is 5, which allows us to see that the value 8.1 is higher than what is typical for this set of data.
Z-score Formula
The Z-score formula is a very useful tool for statisticians. It's given as \( z = \frac{Value - Mean}{Standard\ deviation} \). The Z-score helps us describe where a certain data value lies in relation to the mean of the dataset, in terms of standard deviations.

Let's break down the calculation from the exercise. We have the data value (8.1), the mean (5), and the standard deviation (2). By substituting these into the formula, \( z = \frac{8.1 - 5}{2} \), we find that the data value 8.1 has a Z-score of 1.55.

What Does the Z-score Tell Us?

It tells us that 8.1 is 1.55 standard deviations above the mean. This is quite a bit higher than the average, signaling that 8.1 is not just a random variation but could be significant. If you imagine a bell curve, 8.1 would be sitting to the right of the center.

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Most popular questions from this chapter

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Social jetlag refers to the difference between circadian and social clocks, and is measured as the difference in sleep and wake times between work days and free days. For example, if you sleep between \(11 \mathrm{pm}\) and 7 am on weekdays but from 2 am to 10 am on weekends, then your social jetlag is three hours, or equivalent to flying from the West Coast of the US to the East every Friday and back every Sunday. Numerous studies have shown that social jetlag is detrimental to health. One recent study \(^{64}\) measured the self-reported social jetlag of 145 healthy participants, and found that increased social jetlag was associated with a higher BMI (body mass index), higher cortisol (stress hormone) levels, higher scores on a depression scale, fewer hours of sleep during the week, less physical activity, and a higher resting heart rate. (a) Indicate whether social jetlag has a positive or negative correlation with each variable listed: BMI, cortisol level, depression score, weekday hours of sleep, physical activity, heart rate. (b) Can we conclude that social jetlag causes the adverse effects described in the study?

Exercise 2.143 on page 102 introduces a study that examines the association between playing football. brain size as measured by left hippocampal volume (in \(\mu \mathrm{L}\) ), and percentile on a cognitive reaction test. Figure 2.56 gives two scatterplots. Both have number of years playing football as the explanatory variable while Graph (a) has cognitive percentile as the response variable and Graph (b) has hippocampal volume as the response variable. (a) The two corresponding correlations are -0.465 and \(-0.366 .\) Which correlation goes with which scatterplot? (b) Both correlations are negative. Interpret what this means in terms of football, brain size, and cognitive percentile.

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