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Chapter 7: Problem 96

A typical adult has an average IQ score of 105 with a standard deviation of 20. If 20 randomly selected adults are given an IQ tesst, what is the probability that the sample mean scores will be between 85 and 125 points?

Short Answer

Expert verified
The probability is approximately 1.

Step by step solution

01

Identify the Given Information

We have the population mean (\( \mu \)) as 105 and the population standard deviation (\( \sigma \)) as 20. The sample size (\( n \)) is 20. We are asked to find the probability that the sample mean \( \bar{x} \) is between 85 and 125.
02

Calculate the Standard Error of the Mean

The standard error of the mean (SEM) is calculated using the formula \( \text{SEM} = \frac{\sigma}{\sqrt{n}} \). Substituting the given values, we have:\[ \text{SEM} = \frac{20}{\sqrt{20}} \approx 4.47. \]
03

Convert the Scores to Z-scores

To find the probability, convert the sample mean scores to z-scores using the formula \( z = \frac{\bar{x} - \mu}{\text{SEM}} \).- For \( \bar{x} = 85 \), \[ z = \frac{85 - 105}{4.47} \approx -4.47. \]- For \( \bar{x} = 125 \), \[ z = \frac{125 - 105}{4.47} \approx 4.47. \]
04

Find the Probability Using Z-table

Using a standard normal distribution table (z-table), find the probability corresponding to the z-scores:- For \( z = -4.47 \), the probability is nearly 0.- For \( z = 4.47 \), the probability is nearly 1.The probability that the sample mean is between 85 and 125 is the difference between the probabilities of these z-scores, which is approximately 1 - 0 = 1.

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Key Concepts

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

Probability
Probability is a way to measure the likelihood of an event occurring. When calculating probability, we often use it to determine how likely an event is compared to other possible outcomes. In our exercise, we want to know the probability that the average IQ score of a sample of adults falls within a specific range.

To fully grasp this, imagine flipping a coin. The probability of it landing on heads is 1/2 because there are two equally likely outcomes: heads or tails. Similarly, in the statistics context, we look at the possible outcomes (in this case, IQ scores) and determine the chance of seeing specific values in a given range.
  • Probability range: 0 to 1, where 0 means impossible and 1 means certain.
  • Helps in decision-making based on likely outcomes.
Standard Error of the Mean
The standard error of the mean (SEM) gives us an idea of how much the sample mean is expected to vary from the actual population mean. In simpler terms, it tells us how well a sample's mean is likely to represent the true population mean.

To compute the SEM, we use the formula
\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \]
where \( \sigma \) is the population standard deviation, and \( n \) is the sample size. In the exercise, the SEM helps us understand variability when 20 IQ scores are sampled.
  • Smaller SEM means sample mean is closer to the population mean.
  • Larger sample sizes generally yield smaller SEM.
Z-Score Calculation
Z-scores help us translate raw scores into a standard form, which allows us to compare different data points within a distribution. By converting an IQ score into a z-score, we can determine how far it is from the average population IQ.

The formula to calculate a z-score is:
\[ z = \frac{\bar{x} - \mu}{\text{SEM}} \]
where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, and \( \text{SEM} \) is the standard error of the mean. With these z-scores, we can look up probabilities in a z-table.
  • Positive z-score: above average.
  • Negative z-score: below average.
  • Z-scores near 0 indicate values close to the mean.
Normal Distribution
The normal distribution, also known as the bell curve, is a key concept in statistics. It's a way of showing data that is symmetrically distributed around a mean. Most data points cluster around the mean, with fewer instances as you move away in either direction.

The concept is crucial in our exercise because many natural phenomena, including IQ scores, follow a normal distribution. This makes it easier to make predictions and calculate probabilities with tools like z-scores.
  • Symmetrical shape: mean, median, mode are all equal.
  • Total area under the curve equals 1 (or 100%).
  • Approx. 68% of data falls within 1 standard deviation of mean.

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Most popular questions from this chapter

The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly. Find the sum that is one standard deviation above the mean of the sums.

Richard鈥檚 Furniture Company delivers furniture from 10 A.M. to 2 P.M. continuously and uniformly. We are interested in how long (in hours) past the 10 A.M. start time that individuals wait for their delivery. \(X \sim\) _____(____,____) a. \(U(0,4)\) b. \(U(10,2)\) c. \(E \chi p(2)\) d. \(\quad N(2,1)\)

Suppose in a local Kindergarten through \(12^{\text { th }}\) grade \((\mathrm{K}-12)\) school district, 53 percent of the population favor a charter school for grades \(\mathrm{K}\) through five. A simple rample of 300 is surveyed. Callowing using the normal approximation to the binomial distribtion. a. Find the probability that less than 100 favor a charter school for grades K through 5. b. Find the probability that 170 or more favor a charter school for grades K through 5. c. Find the probability that no more than 140 favor a charter school for grades K through 5. d. Find the probability that there are fewer than 130 that favor a charter school for grades K through 5. e. Find the probability that exactly 150 favor a charter school for grades K through 5.

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