Question

Determine each of the following areas under the standard normal (z) curve: a. To the left of \(-1.28\) b. To the right of \(1.28\) c. Between \(-1\) and 2 d. To the right of 0 e. To the right of \(-5\) f. Between \(-1.6\) and \(2.5\) g. To the left of \(0.23\)

Short answer

a. 0.1003, b. 0.1003, c. 0.8186, d. 0.5, e. Almost 1 (or practically 100%, given that 5 standard deviations from the mean includes almost all possible outcomes), f. 0.9266, g. 0.5910

Step-by-step solution

Map out the Standard Normal Curve

Firstly, visualize a standard normal curve, which is drawn over the x-axis of a graph and is symmetrical. This curve approximately represents the distribution of probability in many natural statistical populations. Having a good understanding of what parts of the curve you're identifying will help solve these problems.

Use a Z-table or Z-score Calculator

To help find the probabilities or areas under the z-curve to the left or right of certain z-scores, use a z-table or a z-score calculator available online.

Solve for Each Part of the Problem

For each lettered part of the problem, locate the given z-score on the z-table or in the z-score calculator and find its corresponding percentile, which will give the area under the curve to the left of the z-score. Areas to the right can be found by subtracting the area to the left from 1.

Write Down the Results

Write down your results as a proportion (out of 1), or convert to a percentage. Don't forget to write down your results in a way that answers each specific part of the problem accurately.

Key concepts

Z-Table

Talking about a z-table takes us into the very heart of statistics, where numbers meet the normal curve. The z-table, to put it simply, is a chart that helps us determine the percentage of values below a certain point in a normal distribution. It's linked to the z-score, which tells us how many standard deviations a value is from the mean.

For instance, if you have a z-score of (-1.28), you would use the z-table to find the area to the left of that z-score on the standard normal curve. This represents the proportion of the data that falls below that value. Reading the z-table might seem daunting with its rows and columns, but it's quite user-friendly once you get the hang of it. You simply find the z-score along the left and top margins and trace down and across to the intersection that gives you your answer.

Z-Score Calculator

A z-score calculator is the modern student's best friend when dealing with the intricacies of standard normal distributions. It's a digital tool that automatically computes the area to the left of a z-score, making it easier for students to understand the implications of their data. Just input your raw score, the mean, and the standard deviation, and the calculator does the rest. It eliminates the potential for human error when reading a z-table and provides results instantly.

Probability Distribution

Understanding Probability Distribution

The basis of statistics is the probability distribution, which describes how the values of a random variable are distributed. It's like a map that shows where the values are most likely to fall. A standard normal distribution, which is just one type of probability distribution, represents data that are symmetrically distributed with a single peak at the mean.

Moreover, the total area under the curve in a probability distribution equals one, as it accounts for all possible outcomes. In terms of percentages, it means a full 100% of the data is accounted for under the curve. This concept becomes particularly significant when we're trying to find the likelihood, or probability, of a certain outcome.

Percentile

What's in a Percentile?

A percentile is a measure that tells us what percentage of the overall dataset falls below a certain value. In the context of a standard normal distribution, it gives us the area to the left of a z-score. For instance, if your score is at the 85th percentile, this means you've scored better than 85% of all the other scores. In exercises that deal with areas under the z-curve, finding the percentile associated with a given data point allows us to make more intuitive sense of the numbers—the higher the percentile, the higher the standing compared to the whole dataset.

Normal Distribution

The Bell Curve Phenomenon

The normal distribution is famously shaped like a bell. It's the graphical representation of data that clusters around a central mean, with values tapering off equally on both sides. This remarkable symmetry signifies that values have a predictable, average behavior with equal likelihood of deviating to the extreme highs or lows. The normal distribution is incredibly useful because many real-world datasets tend to follow this pattern, allowing statisticians to make inferences and predictions about a given population or variable.