Problem 7 Using the Central Limit Theorem ... [FREE SOLUTION]

Chapter 6: Problem 7

Using the Central Limit Theorem assume that females have pulse rates that are normally distributed with a mean of 74.0 beats per minute and a standard deviation of 12.5 beats per minute (based on Data Set 1 "Body Data" in Appendix \(B\) ). a. If 1 adult female is randomly selected, find the probability that her pulse rate is between 72 beats per minute and 76 beats per minute. b. If 4 adult females are randomly selected, find the probability that they have pulse rates with a mean between 72 beats per minute and 76 beats per minute. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed \(30 ?\)

Short Answer

Expert verified

a. 0.1272 b. 0.2510 c. The population distribution is normal.

Step by step solution

Understand the Problem

We need to find the probability that the pulse rate of one adult female or the mean pulse rate of four adult females is between 72 bpm and 76 bpm, given that the distribution is normal with a mean of 74.0 bpm and a standard deviation of 12.5 bpm. And then explain why we can use the normal distribution in part (b) even though the sample size is smaller than 30.

Standardize the Problem for One Female

For part (a), standardize the given pulse rates using the z-score formula \[ z = \frac{X - \mu}{\sigma} \]where \(X\) is the pulse rate (between 72 and 76 bpm), \(\mu\) is the mean (74 bpm), and \(\sigma\) is the standard deviation (12.5 bpm).

Calculate Z-scores for One Female

Calculate the z-scores for 72 bpm and 76 bpm:For 72 bpm:\[ z_1 = \frac{72 - 74}{12.5} = \frac{-2}{12.5} = -0.16 \]For 76 bpm:\[ z_2 = \frac{76 - 74}{12.5} = \frac{2}{12.5} = 0.16 \]

Use Z-tables for Probability for One Female

Using the standard normal distribution table (Z-table), find the probabilities corresponding to \(z = -0.16\) and \(z = 0.16\). For \(z = -0.16\), P(Z < -0.16) 鈮� 0.4364.For \(z = 0.16\), P(Z < 0.16) 鈮� 0.5636.

Calculate Probability for One Female

The probability that the pulse rate is between 72 bpm and 76 bpm is:\[ P(-0.16 < Z < 0.16) = P(Z < 0.16) - P(Z < -0.16) \]\[ 0.5636 - 0.4364 = 0.1272 \]Thus, the probability is approximately 0.1272.

Standardize the Problem for Four Females

For part (b), the sample mean follows a normal distribution with \[ \mu_{\bar{X}} = \mu \]\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]where \(n = 4\). So, \(\sigma_{\bar{X}} = \frac{12.5}{\sqrt{4}} = 6.25\). Standardize using the z-score for \(\bar{X}\): \[ z = \frac{\bar{X} - \mu}{\sigma_{\bar{X}}} \]

Calculate Z-scores for Four Females

Calculate the z-scores for the sample mean of 72 bpm and 76 bpm: For 72 bpm:\[ z_1 = \frac{72 - 74}{6.25} = \frac{-2}{6.25} = -0.32 \]For 76 bpm:\[ z_2 = \frac{76 - 74}{6.25} = \frac{2}{6.25} = 0.32 \]

Use Z-tables for Probability for Four Females

Using the Z-table, find the probabilities corresponding to \(z = -0.32\) and \(z = 0.32\). For \(z = -0.32\), P(Z < -0.32) 鈮� 0.3745.For \(z = 0.32\), P(Z < 0.32) 鈮� 0.6255.

Calculate Probability for Four Females

The probability that the mean pulse rate is between 72 bpm and 76 bpm is:\[ P(-0.32 < Z < 0.32) = P(Z < 0.32) - P(Z < -0.32) \]\[ 0.6255 - 0.3745 = 0.2510 \]Thus, the probability is approximately 0.2510.

Explanation for Part (c)

The Central Limit Theorem states that the distribution of the sample mean will be approximately normal regardless of the sample size, provided the original population distribution is normal. Since the individual pulse rates are normally distributed, we can use the normal distribution for the mean of 4 pulse rates.

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Key Concepts

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

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a bell-shaped curve. Most values cluster around the mean, with fewer values farther from the mean.
It is symmetric, meaning the left and right halves are mirror images. The spread of the curve is determined by the standard deviation.
A smaller standard deviation means the data points are closer to the mean, while a larger standard deviation indicates that they are more spread out. This helps us understand the likelihood of different pulse rates.

Z-Score

A z-score, or standard score, tells us how many standard deviations an element is from the mean. It allows us to compare scores from different distributions.
The formula to calculate a z-score is:
\( z = \frac{X - \text{渭}}{\text{蟽}} \)
Where \(X\) is the raw score (e.g., pulse rate), \(渭\) is the population mean, and \(蟽\) is the standard deviation. This calculation transforms the data into a standard form, allowing us to find probabilities using the standard normal distribution table.
For instance, if a pulse rate is 72 bpm, the z-score helps us understand its position relative to the mean.

Sample Mean

The sample mean is the average value of a sample, representing the central tendency of that sample.
In statistics, when we draw a random sample from a population, the sample mean can be used to estimate the population mean.
The Central Limit Theorem states that the distribution of the sample mean will be approximately normal, regardless of the sample size, provided the original population distribution is normal.
This is why we can calculate probabilities even if our sample size is small, which is very useful in real-world scenarios.
For example, with 4 randomly selected females, we find the sample mean to estimate the probability of their combined pulse rates.

Probability

Probability is a measure of how likely an event is to occur. In statistics, it ranges from 0 (impossible) to 1 (certain).
To find the probability of an event within a normal distribution, we use z-scores and the standard normal distribution table (Z-table).
The Central Limit Theorem allows us to approximate probabilities even for sample means.
By converting raw data to z-scores, we can use Z-tables to determine the probability of a specific range of outcomes, like having a pulse rate between 72 bpm and 76 bpm.
This method helps us answer practical questions about everyday phenomena.

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