(a) Suppose we take a random sample of size 100 from a discrete distribution in this manner: A green die and a red die are thrown simultaneously 100 times and let Xi denote the sum of the spots on the two dice on the ith throw, i = 1, 2,...100. Find the probability that the sample mean number of spots on the two dice is less than 7.5.

(c) A circuit contains three resistors wired in series. Each is rated at 6 ohms. Suppose, however, that the true resistance of each one is a normally distributed random variable with a mean of 6 ohms and a standard deviation of 0.3 ohm. What is the probability that the combined resistance will exceed 19 ohms? How "precise" would the manufacturing process have to be to make the probability less than 0.005 that the combined resistance of the circuit would exceed 19 ohms?

n = 3
µ = 6µ[pic] = 6
σ = 0.3σ[pic] = 0.3 /[pic]

[pic]= 19 / 3

P ([pic]≥ 6.33)= 1- P ( Z < [pic] - µ[pic] )
σ[pic]
= 1 - P (Z < 6.33-6 )
0.3 /[pic]
= 1...

...StandardDeviation
objective
• Describe standarddeviation and
it’s importance in biostatistics.
Measure of Dispersion
• Indicates how widely the scores
are dispersed around the central
point (or mean.)
-StandarddeviationStandardDeviation.
• The most commonly used method
of dispersion in oral hygiene.
• The larger the standarddeviation,
the wider the distribution curve.
StandardDeviation
• SD, , (sigma)
• Indicates how subjects differ from
the average of the group/ the more
they spread out, the larger the
deviation
• Based upon ALL scores, not just
high/low or middle half
• Analyzes descriptively the spread of
scores around the mean
– 14+ 2.51 = Mean of 14 and SD of
2.51
StandardDeviation
• The spread of scores around the
mean:
• For example, if the mean is 60 and
the standarddeviation 10, the
lowest score might be around 30,
and the highest score might be
around 90.
StandardDeviation &
Variance
Usefulness
• When comparing the amount of dispersion in
two data sets.
• Greater variance = greater dispersion
• Standarddeviation--”average” difference
between the mean of a sample and each data
value in the sample
14+ 2.51 = Mean of 14 and SD of 2.51
Distribution...

...I'll be honest. Standarddeviation is a more difficult concept than the others we've covered. And unless you are writing for a specialized, professional audience, you'll probably never use the words "standarddeviation" in a story. But that doesn't mean you should ignore this concept.
The standarddeviation is kind of the "mean of the mean," and often can help you find the story behind the data. To understand this concept, it can help to learn about what statisticians call normal distribution of data.
A normal distribution of data means that most of the examples in a set of data are close to the "average," while relatively few examples tend to one extreme or the other.
Let's say you are writing a story about nutrition. You need to look at people's typical daily calorie consumption. Like most data, the numbers for people's typical consumption probably will turn out to be normally distributed. That is, for most people, their consumption will be close to the mean, while fewer people eat a lot more or a lot less than the mean.
When you think about it, that's just common sense. Not that many people are getting by on a single serving of kelp and rice. Or on eight meals of steak and milkshakes. Most people lie somewhere in between.
If you looked at normally distributed data on a graph, it would look something like this:
The x-axis (the horizontal one) is the value in question......

...StandardDeviation (continued)
L.O.: To find the mean and standarddeviation from a frequency table.
The formula for the standarddeviation of a set of data is [pic]
Recap question
A sample of 60 matchboxes gave the following results for the variable x (the number of matches in a box):
[pic].
Calculate the mean and standarddeviation for x.
Introductory example for finding the mean and standarddeviation for a table:
The table shows the number of children living in a sample of households:
|Number of children, x |Frequency, f |xf |x2f |
|0 |14 |0 × 14 = 0 |02 × 14 = 0 |
|1 |12 |1 × 12 = 12 | |
|2 |8 | | |
|3 |6 | |32 × 6 = 54 |
|TOTAL...

...calls at random to residential phones, both mobile and landline. The purpose of the survey is to determine the percentage of adults who would favor a half-cent increase in the sales tax to help fund public transportation. Four hundred adults are interviewed and 36% of them favor the proposal. Answer the next two questions.
1. The sample size for this sample survey appears to be
a) 400
b) 2,800,000
c) 144
d) 1,008,000
2. The 36% is a
a) Parameter
b) Margin of error
c) Chance of 144 people agreeing to the statement
d) Statistic
3. Event A occurs with probability 0.05. Event B occurs with probability 0.75. If A and B are disjoint, which statement is true?
a) P(A and B) = 0
b) P(A or B) = 0.80
c) P(A and B) = 0.0375
d) Both (a) and (b) are true.
4. Event A occurs with probability 0.05. Event B occurs with probability 0.75. If A and B are independent, which statement is true.
e) P(A and B) = 0
a) P(A or B) = 0.80
b) P(A and B) = 0.0375
c) Both (a) and (b) are true.
A marketing research firm wishes to determine if the adult men in Laramie, Wyoming would be interested in a new upscale men's clothing store. From a list of all residential addresses in Laramie, the firm selects a simple random sample of 100 and mails a brief questionnaire to each. Use this information to answer the next three questions.
5. The population of...

...milligrams of tar per cigarette and a standarddeviation equal to 1.0 milligram. Suppose a sample of 100 low-tar cigarettes is randomly selected from a day’s production and the tar content is measured in each. Assuming that the tobacco company’s claim is true, what is the probability that the mean tar content of the sample is greater than 4.15 milligrams?
[0.00621]
2. The safety limit of a crane is known to be 32 tons. The mean weight and the standarddeviation of a large number of iron rods are 0.3 ton and 0.2 ton respectively. One hundred rods are lifted at a time. Compute the probability of an accident.
[0.1587]
3. A soft –drink vending machine is set so that the amount of drink dispensed is a random variable with a mean of 200 milliliters and a standarddeviation of 15 milliliters. What is the probability that the mean amount dispensed in a random sample of size 36 is at least 204 milliliters?
[0.0548]
4. An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with mean (μ) = 117 cm and standarddeviation (σ) = 6.1 cm. Find the probability that if four subcomponents are randomly selected, their mean length exceeds 120 cm.
[0.16354]
5. The number of pizzas...

...the information provided by the StandardDeviation.
2. The ability to use the StandardDeviation to calculate the percentage of occurrence of a variable either above or below a particular value.
3. The ability to describe a normal distribution as evidenced by a bell shaped curve as well as the ability to prepare a distribution chart from a set of data (module 3 Case).
Part 1
(1) To get the best deal on a CD player, Tom called eight appliance stores and asked the cost of a specific model. The prices he was quoted are listed below:
$ 298 $ 125 $ 511 $ 157 $ 231 $ 230 $ 304 $ 372 Find the Standarddeviation
$ 298 + $ 125+ $ 511+ $ 157+ $ 231+ $ 230+ $ 304+ $ 372= 2228/8 = 278.5(subtract from #s)
19,-153, 232, -121, -47, -48, 25, 93 (square numbers)
380, 2356, 54056, 14762, 2256, 2352, 650, 8742 = 106(added)
(Divide by 7) 15251 (take square root) StandardDeviation = approximately 123.
(2) When investigating times required for drive-through service, the following results (in seconds) were obtained. Find the range, variance, and standarddeviation for each of the two samples, and then compare the two sets of results.
Wendy's 120 123 153 128 124 118 154 110
MacDonald's 115 126 147 156 118 110 145 137
(2) Set 1:
Range : maximum - minimum = 154-110= 44...

...applying to each of the 10 companies and the probability of getting a job offer there. These data are tabulated below. The tabulation is in the decreasing order of cost.
1. If the graduate applies to all 10 companies, what is the probability that she will get at least one offer?
2. If she can apply to only one company, base on cost and success probability criteria alone, should she apply to company 5? Why or why not?
3. If she applies to companies 2,5,8, and 9, what is the total cost? What is the probability that she will get at least one offer?
4. If she wants to be at least 75% confident of getting at least one offer, to which companies should she apply to minimize the total cost?
5. If she is willing to spend $1,500, to which companies should she apply to maximize her chances of getting at least one job?
Company 1 2 3 4 5 6 7 8 9 10
Cost $870 $600 $540 $500 $400 $320 $300 $230 $200 $170
Probability 0.38 0.35 0.28 0.20 0.18 0.18 0.17 0.14 0.14 0.08
A manufacturing company regularly consumes a special type of glue purchased from a foreign supplier. Because the supplier is foreign, the time gap between placing an order and receiving the shipment against that order is long and uncertain. This time gap is called “lead time.” From past experience, the materials manager notes that the company’s demand for glue during the uncertain lead time is normally distributed with a mean of...

...Standarddeviation can be difficult to interpret as a single number on its own. Basically, a small standarddeviation means that the values in a statistical data set are close to the mean of the data set, on average, and a large standarddeviation means that the values in the data set are farther away from the mean, on average.
The standarddeviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standarddeviation.
A small standarddeviation can be a goal in certain situations where the results are restricted, for example, in product manufacturing and quality control. A particular type of car part that has to be 2 centimeters in diameter to fit properly had better not have a very big standarddeviation during the manufacturing process. A big standarddeviation in this case would mean that lots of parts end up in the trash because they don’t fit right; either that or the cars will have problems down the road.
But in situations where you just observe and record data, a large standarddeviation isn’t necessarily a bad thing; it just reflects a large amount of variation in the group that is being studied. For example, if you look at salaries for everyone in a certain company,...

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