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Chapter 7: Problem 33

Find the indicated probability of the standard normal random variable \(Z\). $$P(Z<1.93)$$

Short Answer

Expert verified
\( P(Z < 1.93) = 0.9732 \)

Step by step solution

01

- Understand the Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is often represented by the variable \(Z\).
02

- Identify the Given Z-value

We need to find the probability that the standard normal random variable \(Z\) is less than 1.93, denoted as \(P(Z < 1.93)\).
03

- Use the Z-table

Locate the value 1.93 in the Z-table, which provides the cumulative probability for Z-values. The Z-table shows that \(P(Z < 1.93) = 0.9732\). This means the probability that \(Z\) is less than 1.93 is 0.9732.

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Key Concepts

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

Cumulative Probability
Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain value. In the context of the standard normal distribution, we talk about the probability that the value of the random variable Z is less than or equal to a given Z-value. For example, if we're interested in finding the cumulative probability for Z=1.93, we want to know the probability that Z is less than or equal to 1.93. This cumulative probability is essentially the area under the standard normal curve to the left of the Z-value. Remember, cumulative probabilities increase as Z-values increase.
Z-table
A Z-table is a chart that shows the cumulative probabilities of the standard normal distribution. It helps us find probabilities for specific Z-values. To use the Z-table:
  • Locate the Z-value in the left-most column and top-most row.
  • Combine these to find the cumulative probability.
For instance, in our problem, we need to find the Z-value 1.93. By locating the Z-value 1.9 in the left column and 0.03 in the top row and intersecting them, we find that the cumulative probability is 0.9732. This tells us that the probability that Z is less than 1.93 is 0.9732. Keep in mind that Z-tables typically provide the probability for Z-values less than the chosen value.
Probability Calculation
To calculate the probability of a standard normal random variable Z being less than a certain value, we use the Z-table. Here are the steps:
  • First, identify the Z-value you're interested in. In our case, it is 1.93.
  • Next, use the Z-table to find the cumulative probability corresponding to 1.93.
  • The Z-table tells us that the cumulative probability for Z being less than 1.93 is 0.9732.
Thus, we have calculated that the probability of Z being less than 1.93 is 0.9732, meaning there's a 97.32% chance that Z will fall below 1.93 in a standard normal distribution.

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Most popular questions from this chapter

Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed. A random sample of weekly work logs at an automobile repair station was obtained and the average number of customers per day was recorded. $$\begin{array}{lllll}26 & 24 & 22 & 25 & 23 \\\\\hline 24 & 25 & 23 & 25 & 22 \\\\\hline 21 & 26 & 24 & 23 & 24 \\ \hline 25 & 24 & 25 & 24 & 25 \\\\\hline 26 & 21 & 22 & 24 & 24\end{array}$$

Introduced in the 2000 model year, the Honda Insight was the first hybrid automobile sold in the United States. The mean gas mileage for the model year 2005 Insight with an automatic transmission is 56 miles per gallon on the highway. Suppose the gasoline mileage of this automobile is approximately normally distributed with a standard deviation of 3.2 miles per gallon. (Source: www.fueleconomy.gov) (a) What proportion of 2005 Honda Insights with automatic transmission gets over 60 miles per gallon on the highway? (b) What proportion of 2005 Honda Insights with automatic transmission gets 50 miles per gallon or less on the highway? (c) What proportion of 2005 Honda Insights with automatic transmission gets between 58 and 62 miles per gallon on the highway? (d) What is the probability that a randomly selected 2005 Honda Insight with an automatic transmission gets less than 45 miles per gallon on the highway?

Find the indicated \(Z\) -score. Be sure to draw a standard normal curve that depicts the solution. Find the \(Z\) -score such that the area under the standard normal curve to the right is 0.25

In a Gallup poll conducted December \(2-4,2000,42 \%\) of survey respondents said that, if they only had one child, they would prefer the child to be a boy. Suppose you conduct a survey of 150 randomly selected students on your campus and find that 80 of them would prefer a boy. (a) Approximate the probability that, in a random sample of 150 students, at least 80 would prefer a boy, assuming the true percentage is \(42 \%\) (b) Does this result contradict the Gallup poll? Explain.

Find the indicated probability of the standard normal random variable \(Z\). $$P(-1.20 \leq Z<2.34)$$
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