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Chapter 22: Problem 25

Solve the given problems. With \(75.8 \%\) of the area under the normal curve to the right of \(z\) find the \(z\) -value.

Short Answer

Expert verified
The z-value is approximately -0.70.

Step by step solution

01

Understand the Problem

We are given the percentage of the area under the normal curve to the right of a certain z-value. Our task is to find this z-value when 75.8% of the area is to the right.
02

Convert Percentage to Decimal Probability

Convert the given percentage (75.8%) to a decimal for probability calculations by dividing by 100. Therefore: \(0.758\) is the probability to the right of \(z\).
03

Calculate the Left Probability

The standard normal distribution table (z-table) typically uses the cumulative probability to the left of \(z\). Calculate this by subtracting the right probability from 1. \(1 - 0.758 = 0.242\). The left probability is \(0.242\).
04

Use the z-Table to Find z-Value

Look up or use a standard normal distribution (z-table) to find the z-value corresponding to the cumulative probability of 0.242. This involves checking the z-table for which z-value corresponds to a probability of 0.242.
05

Find the Corresponding z-Value

From the z-table, identify the z-value that gives a cumulative probability of 0.242. This should be a negative z-value since this probability is less than 0.5.

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Key Concepts

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

z-value calculation
Calculating the z-value is the starting point when working with problems involving the normal distribution curve. In simple terms, the z-value鈥攁lso known as the z-score鈥攔epresents the number of standard deviations a data point is from the mean.
Here's how you can calculate it:
  • When you're given a percentage (like 75.8% in this case), you convert it to a decimal by dividing by 100. So, 75.8% becomes 0.758.
  • This value represents the area to the right of the z-value under the normal distribution curve.
  • To work effectively with z-tables鈥攚hich commonly provide data for the left side鈥攜ou'll initially need to compute the corresponding left-side probability.
  • This is done by calculating: \(1 - \text{Right Probability}\). For our example, it鈥檚 \(1 - 0.758 = 0.242\).
The calculation shifts focus from right to left, a crucial step setting you up to consult a z-table.
z-table lookup
Once you've calculated the left-side probability (in our example, 0.242), it's time to use the z-table. The z-table is a statistical tool that helps identify the z-value corresponding to a specific cumulative probability associated with the standard normal distribution curve.
Here鈥檚 how to navigate the z-table:
  • The rows represent the z-value to the tenth place, while the columns provide the hundredth place value.
  • In your search to pinpoint 0.242, start by locating 0.24 in the row section.
  • Scan across this row to find the column number that aligns with our desired probability.
  • The intersection of your row and column gives you the z-value.
Remember, the z-values represent the area to the left under the curve. As our calculated cumulative probability is less than 0.5, the z-value will be negative. Look carefully to ensure you select the correct, corresponding z-value.
cumulative probability
Cumulative probability is central to understanding and interpreting the z-table and the normal curve. Simply put, it refers to the probability that a random variable falls within a certain range or less than a particular value.
In the context of the normal distribution:
  • The cumulative probability indicates the total area under the curve to the left of a z-value.
  • For left-tail probabilities, you use the standard z-table to find the cumulative probability that matches a given z-value.
  • Conversely, for right-side probabilities, you compute cumulative probability by subtracting from 1.
  • Cumulative probability is crucial because it consolidates all probabilities for values less than or equal to a given z-value.
This understanding provides the foundation to look up z-values, interpret distributions, and apply concepts like percentiles and critical values in statistical analyses.

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Most popular questions from this chapter

Solve the given problems. The data in the following table show the percentage of U.S. households that have a landline telephone for various years. Make a time series graph of these data. $$\begin{array}{l|c|c|c|c|c} \text {Year} & 2006 & 2008 & 2010 & 2012 & 2014 \\ \hline \text {Percentage (\%)} & 84 & 78 & 69 & 60 & 53 \end{array}$$

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Use a calculator to find a regression model for the given data. Graph the scatterplot and regression model on the calculator: Use the regression model to make the indicated predictions. The pressure \(p\) at which Freon, a refrigerant, vaporizes for temperature \(T\) is given in the following table. Find a quadratic regression model. Predict the vaporization pressure at \(30^{\circ} \mathrm{F}\). $$\begin{array}{l|c|l|l|l|l}T\left(^{\circ} \mathrm{F}\right) & 0 & 20 & 40 & 60 & 80 \\ \hline p\left(\mathrm{Ib} / \mathrm{in} .^{2}\right) & 23 & 35 & 49 & 68 & 88\end{array}$$

Use the following sets of numbers. \(A: 3,6,4,2,5,4,7,6,3,4,6,4,5,7,3\) \(B: 25,26,23,24,25,28,26,27,23,28,25\) \(C\): 0.48,0.53,0.49,0.45,0.55,0.49,0.47,0.55,0.48,0.57,0.51 0.46,0.53,0.50,0.49,0.53 \(D: 105,108,103,108,106,104,109,104,110,108,108,104\) 113,106,107,106,107,109,105,111,109,108 Determine the mean of the numbers of the given set. Set \(B\)

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. If gas is cooled under conditions of constant volume, it is noted that the pressure falls nearly proportionally as the temperature. If this were to happen until there was no pressure, the theoretical temperature for this case is referred to as absolute zero. In an elementary experiment, the following data were found for pressure and temperature under constant volume. $$\begin{array}{l|c|c|c|c|c|c} T\left(^{\circ} \mathrm{C}\right) & 0.0 & 20 & 40 & 60 & 80 & 100 \\ \hline P(\mathrm{kPa}) & 133 & 143 & 153 & 162 & 172 & 183 \end{array}$$ Use a calculator to find the least-squares line for \(P\) as a function of \(T,\) and from the graph determine the value of absolute zero found in this experiment.
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