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Chapter 6: Problem 6

Do the following: If the requirements of \(n p \geq 5\) and \(n q \geq 5\) are both satisfied, estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution; if \(n p<5\) or \(n q<5\), then state that the normal approximation should not be used. With \(n=8\) births and \(p=0.512\) for a boy, find \(P(\) exactly 5 boys \()\).

Short Answer

Expert verified
The normal approximation should not be used because both np and nq are less than 5.

Step by step solution

01

- Check Requirements

Calculate the values for np and nq to check if the normal approximation can be used. Here, np and nq must both be greater than or equal to 5.Calculate np: \[ n = 8 \quad \text{and} \quad p = 0.512 \]\( np = 8 \times 0.512 = 4.096 \)Calculate nq:\[ q = 1 - p \ q = 1 - 0.512 = 0.488 \]\( nq = 8 \times 0.488 = 3.904 \)Since both np and nq are less than 5 (4.096 < 5 and 3.904 < 5), the normal approximation cannot be used.

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Key Concepts

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

Binomial Distribution
The Binomial Distribution is essential in statistics and probability. It involves scenarios where there are two possible outcomes, often labeled as 'success' and 'failure'. Every trial is independent, and the probability of success remains constant throughout the trials. In our problem, we deal with the probability of having a boy (success) in a series of 8 births. The number of births (n) and the probability of having a boy (p) are the key parameters.
Probability
Probability measures how likely an event is to occur. It ranges from 0 (impossible event) to 1 (certain event). In the context of binomial distribution, we calculate the probability of getting a specific number of successes (e.g., exactly 5 boys in 8 births). However, checking if we meet the conditions for normal approximation simplifies the calculation process by using the normal distribution.
Statistics Education
Understanding these concepts is fundamental in statistics. When learning about binomial and normal distributions, it's crucial to differentiate when to use each method. For the binomial distribution to use the normal approximation, both conditions: \( n p \geq 5 \) and \( n q \geq 5 \) must be satisfied. This ensures the distribution shape is suitable for approximation, making calculations more manageable. Grasping these nuances helps in accurately solving real-world problems using statistics.
Normal Distribution
The Normal Distribution, or the bell curve, is continuous and symmetric around the mean. When used to approximate the binomial distribution, it simplifies complex binomial probability calculations. But it has limitations. In our exercise, the conditions \( n p \) and \( n q \) fail to meet the threshold of 5; thus, the normal approximation isn't suitable. This highlights the importance of verifying conditions before applying statistical methods. Understanding this helps in selecting the correct approach for probability estimations.

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Most popular questions from this chapter

Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Find the probability that a male has a back-to-knee length between \(22.0\) in. and \(24.0\) in.

Assume that females have pulserates 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 鈥淏ody Data鈥� in Appendix B). a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 80 beats per minute. b. If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 80 beats per minute. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed \(30 ?\)

In general, what do the symbols \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) represent? What are the values of \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) for samples of size 64 randomly selected from the population of IQ scores with population mean of 100 and standard deviation of 15 ?

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2. About ________ \(\%\) of the area is between \(z=-2\) and \(z=2\) (or within 2 standard deviations of the mean).

Assume that cans of Coke are filled so that the actual amounts are normally distributed with a mean of \(12.00 \mathrm{oz}\) and a standard deviation of \(0.11\) oz. a. Find the probability that a single can of Coke has at least \(12.19 \mathrm{oz}\). b. The 36 cans of Coke in Data Set 26 "Cola Weights and Volumes" in Appendix \(\mathrm{B}\) have a mean of \(12.19\) oz. Find the probability that 36 random cans of Coke have a mean of at least \(12.19 \mathrm{oz}\) c. Given the result from part (b), is it reasonable to believe that the cans are actually filled with a mean equal to \(12.00\) oz? If the mean is not equal to \(12.00 \mathrm{oz}\), are consumers being cheated?
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