(a) construct a binomial distribution, (b) graph the...

(a) construct a binomial distribution, (b) graph the binomial distribution using a histogram and describe its shape, and (c) identify any values of the random variable $x$ that you would consider unusual. Explain your reasoning. Work Performance Forty-six percent of working mothers say that their work performance is the same as it was before giving birth. You randomly select eight working mothers and ask them how their work performance has changed since giving birth. The random variable represents the number of working mothers who say that their work performance is the same as it was before giving birth.

(a) construct a binomial distribution, (b) graph the binomial distribution using a histogram and describe its shape, and (c) identify any values of the random variable $x$ that you would consider unusual. Explain your reasoning.
Work Performance Forty-six percent of working mothers say that their work performance is the same as it was before giving birth. You randomly select eight working mothers and ask them how their work performance has changed since giving birth. The random variable represents the number of working mothers who say that their work performance is the same as it was before giving birth.

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution used to model the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. It is characterized by two parameters: n (the number of trials) and p (the probability of success in each trial). This distribution is applicable in situations with a yes/no outcome, such as success/failure, and provides the probabilities for obtaining various counts of successes.

Histogram Representation of Discrete Distributions

A histogram is a graphical tool used to represent data distributions by dividing the range of the data into intervals (bins) and displaying the frequency or probability of outcomes within each bin. When graphing a binomial distribution, the histogram displays the probabilities of different counts of successes, which helps in visualizing the shape, spread, and central tendency of the distribution. The shape of the histogram can reveal characteristics such as symmetry or skewness.

Identifying Unusual or Outlier Values

In a probability distribution, unusual values are those that occur with very low probability and are considered outliers. For a binomial distribution, these values typically lie in the tail regions of the distribution, far from the mean. Determining whether a value is unusual involves calculating its probability under the distribution and assessing if it is smaller than a chosen significance level, indicating that such an outcome is unexpected given the distribution's parameters.

Transcript

00:01 In this problem, you're going to construct a binomial distribution, and whenever you are constructing binomial distributions, there are four variables that you have to concern yourself with.

00:16 And those four variables are n, which is the number of trials, x, which is the number of successes, p, which is the probability of success, and q, which is the probability of failure.

00:53 And once you identify all of those variables, then you can apply the formula to calculate the probabilities.

01:01 And the formula is p of x equals ncx, which is the combination of n items taken x at a time, multiplied by the probability of success raised to the x power, multiplied by the probability of failure, multiplied or, sorry, raised to the n minus x power.

01:25 So let's read through our problem and identify the values of all of these variables.

01:33 So this starts with 46 % of working mothers say that their work performance is the same as it was before giving birth.

01:42 And we are going to have the random variable representing the number of working moms who say that their work performance is the same.

01:53 So in this case, our p value is that 46%.

02:02 And if 46 % believe that their work performance is the same as before giving birth, then that means there is 54 % that believe that their work is not the same.

02:18 And we are going to randomly select eight working moms.

02:22 So our n is going to be eight, and therefore our x can be that zero out of those eight in that sample believe their work is the same, maybe only one believes that their work is the same, or two, or three, all the way up to and including eight.

02:45 So we now need to construct a probability distribution, and our probability distribution is going to be a two -consum.

02:54 Column table, listing the possible x values and their associated probabilities.

03:17 Now when it comes time to calculate those probabilities, we're going to use our n value, and our x value is going to continue to change.

03:32 The first go round, it's going to be a zero, the second go round it's going to be a one and so forth.

03:37 We're going to multiply that by the p value of 0 .46 raised to the x power and then we're going to multiply it by the q value 0 .5 4 raised to the 8 minus x value now the most efficient way to tackle this is to utilize your technology and your graphing calculator so we're going to bring in our graphing calculator and if there's anything in your lists you do want to clear them out and to access lists we're going to hit our stat button and edit and in list one we're going to type in the zero through eight and then we're going to sit on top of list one or sorry top of list two and we're going to start typing in the formula with all the variable values put in but because we've just housed all of our xes in list one everywhere we see an x, we're going to type in list 1.

05:00 So we're going to have 8.

05:02 We're going to access our combination function under our math button.

05:07 And we're going to scoot over to the probability section and access our combination function.

05:14 And then we're going to type in list 1.

05:16 So we're going to hit the second button and the 1 button.

05:21 We're going to multiply that by the p value, which is 0 .46, raised 2 .3.

05:26 Instead of x, we're going to type in list 1.

05:30 And we're going to multiply that by our q value, which is 0 .54, raised to the power of n, which in this case was 8, minus x, which in this case we're going to type in list 1.

05:46 And we're going to hit enter, and it's going to complete our entire distribution.

05:52 And then we're going to just copy everything over.

05:55 So we are going to copy over .00723...

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