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An office supply company conducted a survey before marketing a new paper shredder designed for home use. In the survey, \(80 \%\) of the people who used the shredder were satisfied with it. Because of this high acceptance rate, the company decided to market the new shredder. Assume that \(80 \%\) of all people who will use it will be satisfied. On a certain day, seven customers bought this shredder. a. Let \(x\) denote the number of customers in this sample of seven who will be satisfied with this shredder. Using the binomial probabilities table (Table I, Appendix C), obtain the probability distribution of \(x\) and draw a graph of the probability distribution. Find the mean and standard deviation of \(x\). b. Using the probability distribution of part a, find the probability that exactly four of the seven customers will be satisfied.

Step by step solution

01

Find the binomial probabilities

The probability of success (a customer being satisfied) is given as \(0.8\), and failure (not being satisfied) as \(0.2\). With 7 customers, \(x\) can be any number from 0 to 7. So, calculate each binomial probability \(P(X=x) = ^7C_x *(0.8)^x * (0.2)^{7-x}\) where \(^7C_x\) stands for the number of combinations of 7 items taken x at a time.

02

Draw the probability distribution

On the x-axis, plot the different values of \(x\) (from 0 to 7) and on the y-axis, plot their corresponding probability calculated in Step 1. The graph will show the probability distribution.

03

Calculate the mean and standard deviation

The mean (\(µ\)) and standard deviation (\(σ\)) of a binomial distribution can be found using the formulas \(µ = np\) and \(σ = \sqrt{np(1-p)}\) where \(n=7\) is the number of trials and \(p = 0.8\) is the probability of success.

04

Find the probability of exactly 4 satisfied customers

Use the binomial probability formula \(P(X=4) = ^7C_4 *(0.8)^4 * (0.2)^{7-4}\)

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Key Concepts

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

Understanding Probability Distribution

In statistics, a probability distribution tells us how likely different potential outcomes are. Imagine rolling a die – each side (1 through 6) is an outcome with an equal chance of occurring. Similarly, in the shredder example, we have outcomes ranging from "0 customers satisfied" to "7 customers satisfied." Each outcome has its own probability.

When we calculate these probabilities using specific formulas (like the binomial probability formula), we get a complete probability distribution. For the shredder company, knowing these probabilities helps paint a clear picture of customer satisfaction outcomes. We can visualize this information on a graph, where the x-axis represents the number of satisfied customers and the y-axis represents the probability of each outcome.

Mean and Standard Deviation Explained

The mean and standard deviation are essential tools for understanding data in binomial distributions. The mean (\(µ\)) gives us an average – it tells us what we can typically expect. For example, with a satisfaction probability of 80% and seven customers, we can calculate:

• Mean (\(\mu\)): \(np = 7 * 0.8 = 5.6\)

This tells us that, on average, about 5.6 out of 7 customers are satisfied. But, since we can't have a fraction of a customer, it suggests around 5 or 6 satisfied customers most of the time.

The standard deviation (\(\sigma\)) shows the variability – how spread out these satisfaction outcomes can be. Using the formula \(\sigma = \sqrt{np(1-p)}\), we find:

• Standard Deviation: \(\sqrt{7*0.8*0.2} \approx 1.17\)

This value tells us that while 5.6 is the average, actual results typically vary by about 1 satisfied customer this way or that.

Using the Binomial Probability Formula

The binomial probability formula is your best friend in calculating specific outcome probabilities in situations like this. You use it when you know the number of trials (in this case, 7 customers) and the success probability (here, 0.8 satisfaction rate). The formula is:

• \[P(X=x) = {^nC_x} * p^x * (1-p)^{n-x}\] where nC_x is the number of combinations.

It's essential for part b of our exercise, where we need to know the exact probability of "exactly four satisfied customers." Plug in the values:

• \(P(X=4) = {^7C_4} * (0.8)^4 * (0.2)^3\)

By solving this, you'll get the precise probability of exactly four satisfied customers. Mastering this formula is key to understanding how likely specific outcomes are.

Evaluating Satisfaction Rate

In business, the satisfaction rate can be a powerful metric. It represents the proportion of successful outcomes. Here, it's the percentage of customers satisfied with the shredder. The survey showed an 80% satisfaction rate, which the company uses as its basis for decision-making.

This high satisfaction rate with the shredder can be advantageous in marketing and sales. It gives confidence in product performance and sets expectations for future customer satisfaction. Understanding and using satisfaction rates can enhance marketing strategies and customer relationship management. For any product, knowing the satisfaction rate helps predict customer happiness and potentially alerts us to areas that need improvement if the rate isn't high.