Question

Did you ever buy an incandescent light bulb that failed (either burned out or did not work) the first time you turned the light switch on? When you put a new bulb into a light fixture, you expect it to light, and most of the time it does. Consider 8 -packs of 60 -watt bulbs and let $x$ be the number of bulbs in a pack that "fail" the first time they are used. If 0.02 of all bulbs of this type fail on their first use and each 8 -pack is considered a random sample, a. List the probability distribution and draw the histogram of $x.$ b. What is the probability that any one 8 -pack has no bulbs that fail on first use? c. What is the probability that any one 8 -pack has no more than one bulb that fails on first use? d. Find the mean and standard deviation of $x .$ e. What proportion of the distribution is between $\mu-\sigma$ and $\mu+\sigma ?$ f. What proportion of the distribution is between $\mu-2 \sigma$ and $\mu+2 \sigma ?$ g. How does this information relate to the empirical rule and Chebyshev's theorem? Explain. h. Use a computer to simulate testing 1008 -packs of bulbs and observing $x,$ the number of failures per 8 -pack. Describe how the information from the simulation compares with what was expected (answers to parts a-g describe the expected results). i. $\quad$ Repeat part h several times. Describe how these results compare with those of parts a-g and with part h.

Name: Problem 96, Chapter 5: Elementary Statistics
Uploaded: 2020-06-18T02:29:51-08:00
Duration: 12 min 32 s
Channel: Bon Zapata

   Did you ever buy an incandescent light bulb that failed (either burned out or did not work) the first time you turned the light switch on? When you put a new bulb into a light fixture, you expect it to light, and most of the time it does. Consider 8 -packs of 60 -watt bulbs and let $x$ be the number of bulbs in a pack that "fail" the first time they are used. If 0.02 of all bulbs of this type fail on their first use and each 8 -pack is considered a random sample,
a. List the probability distribution and draw the histogram of $x.$
b. What is the probability that any one 8 -pack has no bulbs that fail on first use?
c. What is the probability that any one 8 -pack has no more than one bulb that fails on first use?
d. Find the mean and standard deviation of $x .$
e. What proportion of the distribution is between $\mu-\sigma$ and $\mu+\sigma ?$
f. What proportion of the distribution is between $\mu-2 \sigma$ and $\mu+2 \sigma ?$
g. How does this information relate to the empirical rule and Chebyshev's theorem? Explain.
h. Use a computer to simulate testing 1008 -packs of bulbs and observing $x,$ the number of failures per 8 -pack. Describe how the information from the simulation compares with what was expected (answers to parts a-g describe the expected results).
i. $\quad$ Repeat part h several times. Describe how these results compare with those of parts a-g and with part h.

Elementary Statistics

Robert Johnson,… 11th Edition

Chapter 5, Problem 96

Instant Answer

Solved by Expert Bon Zapata

06/18/2020

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02 chance of failing, and each box contains 8 light bulbs. We are asked to find the probability distribution. This is a binomial distribution problem, where $n=8$ (the number of trials) and $p=0.02$ (the probability of success on each trial). The binomial Show more…

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Did you ever buy an incandescent light bulb that failed (either burned out or did not work) the first time you turned the light switch on? When you put a new bulb into a light fixture, you expect it to light, and most of the time it does. Consider 8 -packs of 60 -watt bulbs and let $x$ be the number of bulbs in a pack that "fail" the first time they are used. If 0.02 of all bulbs of this type fail on their first use and each 8 -pack is considered a random sample, a. List the probability distribution and draw the histogram of $x.$ b. What is the probability that any one 8 -pack has no bulbs that fail on first use? c. What is the probability that any one 8 -pack has no more than one bulb that fails on first use? d. Find the mean and standard deviation of $x .$ e. What proportion of the distribution is between $\mu-\sigma$ and $\mu+\sigma ?$ f. What proportion of the distribution is between $\mu-2 \sigma$ and $\mu+2 \sigma ?$ g. How does this information relate to the empirical rule and Chebyshev's theorem? Explain. h. Use a computer to simulate testing 1008 -packs of bulbs and observing $x,$ the number of failures per 8 -pack. Describe how the information from the simulation compares with what was expected (answers to parts a-g describe the expected results). i. $\quad$ Repeat part h several times. Describe how these results compare with those of parts a-g and with part h.

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Key Concepts

Simulation in Statistical Analysis

Simulation involves using computational methods to mimic the behavior of a complex system or process. In statistics, simulations can be used to approximate the behavior of random experiments, validate theoretical probability distributions, and understand variability in outcomes. By repeatedly sampling or replicating a process, one can gather empirical evidence about the expected distribution of outcomes, which can then be compared to analytical predictions to check their accuracy or gain further insights.

Empirical Rule and Chebyshev's Theorem

The empirical rule and Chebyshev's theorem are tools used to understand the dispersion of data about the mean. The empirical rule, applicable to normally distributed data, states that approximately 68%, 95%, and 99.7% of the data fall within one, two, and three standard deviations of the mean, respectively. Chebyshev's theorem, on the other hand, applies to any distribution regardless of shape and provides a conservative estimate of the minimum proportion of values that lie within k standard deviations from the mean. Comparing these two concepts helps in deciding which method is more appropriate based on the underlying distribution of the data.

Probability Distribution and Histogram

A probability distribution describes how the probability of an event or outcome is distributed over the set of all possible outcomes. In the context of discrete distributions like the binomial, a histogram can be used as a visual tool to represent these probabilities by displaying data with bars where the height of each bar corresponds to the probability or frequency of that outcome. This helps in understanding the shape and spread of the distribution.

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent 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 on each trial. This distribution is especially useful when the outcome of each trial can only result in success or failure, allowing one to calculate probabilities for various counts of successes based on these parameters.

Mean and Standard Deviation of a Binomial Distribution

The mean (or expected value) and standard deviation are measures of central tendency and spread for a distribution, respectively. For a binomial distribution, the mean is calculated as np, where n is the number of trials and p is the probability of success. The standard deviation is given by the square root of np(1-p). These formulas help summarize the distribution by providing insight into the average outcome and the variability around that average.

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Transcript

00:01 So we're given that a light bulb has a 0 .02 chance of failing, and each box has eight light bulbs in it.

00:08 So the first thing we're going to do is find the probability distribution.

00:13 Now, on this page and on the last page of the buck, you are given a lot of notes about how to do these problems on a computer.

00:21 That's because they want you to do these problems on either a computer or graphing calculator.

00:27 I'm going to be doing this in excel.

00:30 Well, you can do this in other programs.

00:32 The steps might be a little bit different, though.

00:37 So if we do this in excel, first thing i'm going to do is make a list of all the possible outcomes.

00:44 There's eight trials.

00:46 So we need the number 0 to 8.

00:50 Let's do this.

00:52 Select all of them and drag down.

00:55 Then i'm going to go into b1.

00:57 And under formulas and statistical, and i use the binomial distribution tool.

01:07 So our trials, that's eight because there's eight light bulbs in the box.

01:12 Our probability was 0 .02.

01:15 We don't have to convert it like we normally do because we weren't given it as a percentage.

01:20 It's already in the form we want it to be in.

01:25 Cumulative is false.

01:26 We're not working with this right now.

01:28 So for this, the number, i'm going to put all my numbers from zero to eight.

01:34 So starting with the a1 box, colon the a9 box.

01:41 And we get this.

01:45 So now we need to draw a histogram of x.

01:51 So for a histogram of x, we put everything in order.

01:57 And our x values are going to be the x axis.

02:01 So that's just going to be the numbers from 0 to 8.

02:06 So 2, 3, 4, 5, 6, 7, and 8.

02:15 And on our y -axis is going to be all of the probabilities.

02:23 However, let's go back and take a look here.

02:27 You notice the first one's the biggest, then the second one's smaller, smaller, and it keeps getting smaller and smaller.

02:34 This e means times 10 to the blank power.

02:40 So, for example, the probability of 7 bolt being bad is 1 times 10 to the negative below power.

02:49 So the reason why it keeps getting smaller is because of this probability is so low that getting zero or 1, that's the most realistic outcome.

02:58 So we can make our histogram like a descending staircase kind of.

03:05 So zero is going to be the highest.

03:09 Then one, then two, then three, four.

03:14 And eventually these ones are going to get so small that it's just a tiny little dash.

03:20 And then we'll fill in some values.

03:22 So the one for zero, that was a 0 .851.

03:27 The one at 1, that was .139.

03:36 Then we have .0099.

03:42 And i'll put one more.

03:44 The one at 3, we have 0 .0004.

03:51 So this is a decent histogram.

03:57 So what is the probability of getting a box with no failing light bulbs? so let's go back to our distribution that we just made.

04:06 So no failing light bulbs, that's zero.

04:10 So that's right here, the 0 .81.

04:14 This problem is just reading the table.

04:19 So 0 .851, and that's because the no here means zero.

04:25 What is the probability of getting a box with no more than one? so what we'll do for this one is we'll take all the possible options.

04:36 No more than one means either zero.

04:38 Or 1.

04:41 So we'll take both of these two and add them together.

04:46 And when you do that, you should get 0 .999.

04:52 We just add the first two together.

04:58 Find the mean and standard deviation of x.

05:01 So because we have a binomial random variable, either the light bulb fails or it works, so it's binomial.

05:08 These are the two equations we use to get mean and standard deviation.

05:14 Your notes might say a q here instead of 1 minus p.

05:18 That's the same thing.

05:21 So n is our number of trials.

05:24 That is 8.

05:26 And p is our probability.

05:28 That's 0 .02...

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