Refer to Exercise 4.70 (p. 240 ) and the associated poll...

Refer to Exercise 4.70 (p. 240 ) and the associated poll that revealed that half of all pet owners would get their next dog or cat from a shelter. Consider a random sample of 700 pet owners and define $x$ as the number of pet owners who would acquire their next dog or cat from a shelter. In Exercise $4.70,$ you determined that $x$ is a binomial random variable. a. Compute $\mu$ and $\sigma$ for the probability distribution of $x$ b. Compute the $z$ -score for the value $x=338$. c. Compute the $z$ -score for the value $x=374$. d. Use the technique of this section to approximate $P(338<x<372)$

Refer to Exercise 4.70 (p. 240 ) and the associated poll that revealed that half of all pet owners would get their next dog or cat from a shelter. Consider a random sample of 700 pet owners and define $x$ as the number of pet owners who would acquire their next dog or cat from a shelter. In Exercise $4.70,$ you determined that $x$ is a binomial random variable.
a. Compute $\mu$ and $\sigma$ for the probability distribution of $x$
b. Compute the $z$ -score for the value $x=338$.
c. Compute the $z$ -score for the value $x=374$.
d. Use the technique of this section to approximate $P(338<x<372)$

Key Concepts

Binomial Distribution

A binomial distribution is a probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes—success or failure—with a constant probability of success. It is defined by the parameters n (the number of trials) and p (the probability of success on each trial).

Mean of a Binomial Distribution

The mean (expected value) of a binomial distribution is given by the formula ? = n * p. This value represents the average number of successes expected over many repetitions of the experiment.

Standard Deviation of a Binomial Distribution

The standard deviation of a binomial distribution is calculated using the formula ? = ?(n * p * (1-p)). It measures the dispersion of the number of successes around the mean, providing an understanding of the variability inherent in the process.

Z-Score Calculation

A z-score is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. It is computed by subtracting the mean from the observed value and dividing the result by the standard deviation. This standardization enables comparison between different data points or different distributions.

Normal Approximation to a Binomial Distribution

When the number of trials n is large and the probability p is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution. This is due to the central limit theorem, which states that the distribution of the sum (or average) of a large number of independent, identically distributed variables tends to be normal, regardless of the original distribution.

Continuity Correction

Continuity correction is an adjustment made when a discrete probability distribution (such as the binomial) is approximated by a continuous distribution (such as the normal). This correction involves adding or subtracting 0.5 to the discrete x-values in order to improve the approximation accuracy. It accounts for the fact that a continuous distribution covers a continuous range of values while a discrete distribution does not.

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