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Chapter 6: Problem 7

Find the area under the normal curve that lies to the left of the following \(z\)-values. a. \(\quad z=-1.30\) b. \(\quad z=-2.56\) c. \(\quad z=-3.20\) d. \(\quad z=-0.64\)

Short Answer

Expert verified
The areas under the standard normal curve to the left of the given z-values are approximately: a) 0.0968, b) 0.0052, c) 0.0007, d) 0.2607. Remember that these are approximate values and might differ slightly depending on the precision of the z-table used.

Step by step solution

01

Understanding the z-table

Recognize that a z-table provides the area to the left of a given z-value under the standard normal curve. The table is looked up with the z-value. For negative z-values, the area to the left of the z is given directly in the table.
02

Lookup z=-1.30

Look up the value for -1.30 in the z-table, navigating to the row for -1.3, and then moving across to the column labelled '.00' which gives the area to the left of z=-1.3.
03

Lookup z=-2.56

Next, the value for -2.56 is looked up in the z-table, choosing the row for -2.5 and then the column for '.06'.
04

Lookup z=-3.20

For z=-3.20, likewise, find the row for -3.2 and the column for '.00' to get the area to the left of z=-3.2.
05

Lookup z=-0.64

Finally, look up the value for -0.64 in the z-table, navigating to the row for -0.6 and then the column for '.04'.

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Key Concepts

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

Understanding the Z-table
A z-table is a powerful tool used in statistics to find the probability of a specific z-value within a standard normal distribution. Essentially, this table provides the area under the curve to the left of a given z-value. For the normal distribution, which is symmetrical, the z-table helps us determine how much of the data falls below a certain z-score. To use the z-table:
  • Identify the z-value you are interested in.
  • Locate the z-value in the table, which is usually divided into rows and columns, with rows representing whole numbers and first decimal places, while columns represent second decimal places.
  • Read off the number at the intersection, which represents the area to the left of that z-value.
For example, if you need to find the value for a z-score of -1.30, locate the row for -1.3, then find the column for '.00', and the intersecting value will tell you the cumulative probability.
Area Under the Curve
The area under the curve in a normal distribution represents probabilities and is one of the fundamental concepts in statistics. Probability values range from 0 to 1, where:
  • An area of 0 means there is no probability of the event.
  • An area of 1 indicates certainty of the event happening.
In the context of a standard normal distribution, which is a bell-shaped curve, the total area under the curve is always equal to 1. This makes finding probabilities simpler. When tasked with finding the area to the left of a specific z-value, you're essentially calculating the cumulative probability, i.e., the probability of a random variable being less than the z-value.
The z-table often provides this exact left-ward cumulative probability, hence why it's a vital part of this calculation.
Grasping Z-values
Z-values, also known as z-scores, are standardized scores that convert data into terms of how many standard deviations away from the mean they are. This conversion to z-scores is vital because it allows comparison across different data sets.Here's a simple overview of calculating z-scores:
  • The formula: \( z = \frac{x - \mu}{\sigma} \), where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
  • A positive z-value means the data point is above the mean.
  • A negative z-value indicates it is below the mean.
Once z-values are calculated, they can easily be utilized in a z-table to find the necessary probabilities, as discussed previously.
The Standard Normal Curve
The standard normal curve, often associated with the Gaussian distribution, is a key concept in statistics. It is characterized by a mean of 0 and a standard deviation of 1, which forms the basis for the z-scores. Its shape is symmetrical, forming the characteristic bell-shaped curve. This structure is important because it makes calculating and interpreting data straightforward:
  • The peak of the curve corresponds to the mean, median, and mode, all being at the center.
  • The tails of the curve extend infinitely while never touching the axis, which indicates the variability of data.
  • Many natural phenomena fit this normal distribution, leading to its widespread usage in statistical analysis and probability.
Using z-tables alongside the standard normal curve allows for the easy computation of probabilities and understanding of how data is spread across this distribution.

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Most popular questions from this chapter

Suppose you were to generate several random samples, all the same size, all from the same normal probability distribution. Will they all be the same? How will they differ? By how much will they differ? a. Use a computer or calculator to generate 10 different samples, all of size \(100,\) all from the normal probability distribution of mean 200 and standard deviation 25. b. Draw histograms of all 10 samples using the same class boundaries. c. Calculate several descriptive statistics for all 10 samples, separately. d. Comment on the similarities and the differences you see.

According to ACT, results from the 2008 ACT testing found that students had a mean reading score of 21.4 with a standard deviation of \(6.0 .\) Assuming that the scores are normally distributed, a. find the probability that a randomly selected student had a reading ACT score less than \(20 .\) b. find the probability that a randomly selected student had a reading ACT score between 18 and 24 c. find the probability that a randomly selected student had a reading ACT score greater than 30 d. find the value of the 75th percentile for ACT scores.

Find the standard score \(z\) such that the area below the mean and above \(z\) under the normal curve is a. 0.3212 b. 0.4788. c. 0.2700.

According to Chebyshev's theorem, at least how much area is there under the standard normal distribution between \(z=-2\) and \(z=+2 ?\) What is the actual area under the standard normal distribution between \(z=-2\) and \(z=+2 ?\)

A soft drink vending machine can be regulated so that it dispenses an average of \(\mu\) oz of soft drink per cup. a. If the ounces dispensed per cup are normally distributed with a standard deviation of 0.2 oz, find the setting for \(\mu\) that will allow a 6 -oz glass to hold (without overflowing) the amount dispensed \(99 \%\) of the time. b. Use a computer or calculator to simulate drawing a sample of 40 cups of soft drink from the machine (set using your answer to part a).
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