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Chapter 7: Problem 14

A normal population has a mean of 12.2 and a standard deviation of 2.5 a. Compute the \(z\) value associated with 14.3 . b. What proportion of the population is between 12.2 and \(14.3 ?\) c. What proportion of the population is less than \(10.0 ?\)

Short Answer

Expert verified
a. z = 0.84; b. 29.95% of the population is between 12.2 and 14.3; c. 18.94% is less than 10.0.

Step by step solution

01

Understand the formula for the z-value

The formula for computing the z-value is: \[ z = \frac{X - \mu}{\sigma} \]where \(X\) is the value for which you are finding the z-score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
02

Calculate the z-value for 14.3

Here, \(X = 14.3\), \(\mu = 12.2\), and \(\sigma = 2.5\). Substitute these into the z-score formula: \[ z = \frac{14.3 - 12.2}{2.5} = \frac{2.1}{2.5} = 0.84 \]So, the z-value associated with 14.3 is 0.84.
03

Find the proportion of the population between the mean and 14.3

To find this proportion, we refer to the z-table and look up the z-value of 0.84. The z-table shows the area to the left of a given z-score, but since we need the area between the mean (z=0) and z=0.84, we find the area under the curve for z=0.84. Typically, this is about 0.2995 or 29.95% of the population.
04

Calculate the proportion of the population below 10.0

First, compute the z-value for 10.0 using the formula:\[ z = \frac{10.0 - 12.2}{2.5} = \frac{-2.2}{2.5} = -0.88 \]Now, refer to the z-table to find the area to the left of z=-0.88. This area represents the proportion of the population below 10.0, which is approximately 0.1894 or 18.94%.

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Key Concepts

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

Normal Distribution
Understanding normal distribution is like getting to know a bell-shaped curve. This curve is symmetrical and illustrates how data points are distributed across a population. In most cases, data near the mean are more frequent in occurrence than data far from the mean. Here are some key points to remember:
  • The curve is centered around the mean, which is the average value, in this case, 12.2.
  • Most values cluster around the mean, tapering off as they move away.
  • Normal distribution is used in statistics to represent real-valued random variables with unknown distributions.
  • It provides a basis for several statistical analyses, including the z-score calculations.
Overall, knowing the characteristics of normal distribution can help in understanding the distribution of values like in the case of computing proportions within a given range, such as between 12.2 and 14.3.
Standard Deviation
Standard deviation measures the amount of variation or dispersion in a set of values. In our problem, the standard deviation is 2.5, indicating how spread out the numbers are from the average. Here's a closer look at what standard deviation tells us:
  • A low standard deviation means data points are close to the mean.
  • A high standard deviation indicates data points are spread out over a larger range.
  • It's a crucial component of the z-score formula, helping determine how far away a point is from the mean.
Think of standard deviation like a "how spread out" measure. In practical terms, if an event is one standard deviation away from the mean, it follows the pattern set by the normal distribution curve. For instance, the z-score calculation for 14.3 tells us it is 0.84 standard deviations away from 12.2, indicating it is relatively close to the average but slightly above.
Proportion of Population
The proportion of the population refers to the fraction of the total individuals that fall into a certain category. To find this using a z-score you proportionally relate areas under the curve in a normal distribution. Here's how the concept works in practice:
  • You determine the range of interest (e.g., values between the mean and 14.3).
  • Compute the z-score for these values to find out how many standard deviations they are from the mean.
  • Use a z-table to find the corresponding area under the curve, which gives you the proportion of the population within that range.
For instance, the exercise finds approximately 29.95% of the population falls between 12.2 and 14.3. Similarly, a z-table finds that about 18.94% falls below 10.0. These proportions help in making informed decisions or predictions based on statistical data, whether in academics or real-world scenarios.

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Most popular questions from this chapter

Explain what is meant by this statement: "There is not just one normal probability distribution but a 'family' of them."

Refer to the CIA data, which report demographic and economic information on 46 countries. a. The mean of the GDP/capita variable is 16.58 with a standard deviation of \(9.27 .\) Use the normal distribution to estimate the percentage of countries with exports above 24. Compare this estimate with the actual proportion. Does the normal distribution appear accurate in this case? Explain. b. The mean of the exports is 116.3 with a standard deviation of 157.4 . Use the normal distribution to estimate the percentage of countries with exports above 170\. Compare this estimate with the actual proportion. Does the normal distribution appear accurate in this case? Explain.

The weights of canned hams processed at Henline Ham Company follow the normal distribution, with a mean of 9.20 pounds and a standard deviation of 0.25 pounds. The label weight is given as 9.00 pounds. a. What proportion of the hams actually weigh less than the amount claimed on the label? b. The owner, Glen Henline, is considering two proposals to reduce the proportion of hams below label weight. He can increase the mean weight to 9.25 and leave the standard deviation the same, or he can leave the mean weight at 9.20 and reduce the standard deviation from 0.25 pounds to \(0.15 .\) Which change would you recommend?

The time patrons at the Grande Dunes Hotel in the Bahamas spend waiting for an elevator follows a uniform distribution between 0 and 3.5 minutes. a. Show that the area under the curve is 1.00 . b. How long does the typical patron wait for elevator service? c. What is the standard deviation of the waiting time? d. What percent of the patrons wait for less than a minute? e. What percent of the patrons wait more than 2 minutes?

A uniform distribution is defined over the interval from 6 to 10 a. What are the values for \(a\) and \(b\) ? b. What is the mean of this uniform distribution? c. What is the standard deviation? d. Show that the total area is 1.00 . e. Find the probability of a value more than 7 . f. Find the probability of a value between 7 and \(9 .\)
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