statistics Midterm Chapters 1-6

Calculate the z-score of the given X value, X = 46.2, where mean = 48.2, and S.D. = 47.7.

(46.2-48.2)/47.7 = -0.04 Correct Answer -0.04 CH. 6.1

Calculate the standard score of the given X value, X=64.1, where mean=72.6, and S.D.=68.1, and indicate where z will be located.

(X-mean)/S.D. = z-score (64.1 - 72.6) / 68.1 = -0.12 CH. 6

The life of light bulbs is distributed normally. The variance of the life time is 625 and the mean lifetime of a bulb is 560 hours. Find the probability of a bulb lasting for at most 602 hours.

A find the Z-Score problem. Then use the Z to find Area. Z = (602 - 560) / 25 = 1.68 = Z Using Table, find the area: Z-1.68 equals 0.9535 Correct Answer is 0.9535 CH. 6.3

Meta-analysis or a case study: An underwriter says the risk of fire destroying a home is high in a certain area due to the lack of fire hydrants in the neighborhood or a fire station nearby.

Case Study CH. 1

Identify the sampling technique: For budget purposes, a college president needs to know the average cost of health plans of instructors at their college>

Census CH. 1

When Order in NOT important

Combination nCr=n!/r!(n-r)! CH. 4

Speeds of cars driving on the highway are an example of which type of data? Discrete, Continuous, or neither?

Continuous [variable] can assume all values between any two specific values by measuring

Descriptive or inferential statistic?: A recent poll of 2907 home owners in W.V. showed that the average price of a house in the U.S. is $276, 500.

Descriptive: Organizing, summarizing, presentation of data. ANSWER: Inferential: Consists of making inferences from samples to populations. CH. 1

A subset of outcomes of the sample space is called a(n) ______.

Event CH. 4

Find the area under the normal curve between z = -2.72 and z = 1.5.

Look up the area to the left of Z1 = -2.72, which is 0.0033 Look up the area to the left of Z2 = 1.5, which is 0.9332 Z2 - Z1 = 0.9332 - 0.0033 = 0.9299 Answer is 0.9299 CH. 6.2

is -0.38 a probability?

No. Because probabilities must be between zero and one, inclusively. 0<=p<=1 CH. 4

A coordinator will select 4 songs from a list of 6 songs to compose. How may different lineups are possible?

ORDER IS IMPORTANT! Permutation nPr=n!/(n-r)!

When Order Is Important

Permutation nPr=n!/(n-r)! CH. 4

Qualitative or Quantitative? Discrete or Continuous? Highest level of measurement?: A questionnaire asked the respondent to identify his/her gender. The respondent is asked to fill the appropriate number on a form. 1) Male 2) Female

Qualitative Neither Discrete or Continuous Discrete-values can be counted/Continuos- Nominal

Qualitative or Quantitative? Discrete or Continuous? Highest level of measurement?: The number of days traveled each month by 100 randomly selected employees.

Quantitative-can be ordered or ranked. Discrete-values can be counted Ratio- CH. 1

_____ occurs if a researcher unintentionally influences the subject responses by their actions.

Researcher Bias CH. 1

Probability of rolling a number less than 4 with a standard six-sided die.

Rolling a 1, 2, or 3. 3/6=1/2=0.5000. CH. 4

A statistics student chooses twenty people at random from each class

Stratified Sampling CH. 1

Frequency Table Class Width

Subtract the lower limit of one class and the lower limit of the next class. CH. 2

Cumulative Frequency

To find a cumulative frequency, sum the frequencies of the class given and all previous classes.

Create the probability distribution for "The number of heads in 4 tosses of a coin."

X 0 1 2 3 4 P(X=x) 1/16 1/4 3/8 1/4 1/16 CH. 5

Midpoint equals

lower boundary + upper boundary divided by two. CH. 2

What is the probability of drawing a club, and w/o replacement, a black card.

standard deck has 13 clubs and 26 black cards. Club: 13/52 and Black Card: 25/51 13/52 x 25/51 = 0.1225 CH. 4

What is the sample space for 2 coins tossed. H-heads and T-tails.

{HH,HT,TH,TT} CH. 4

Diameter of ball bearing are distrusted normally. The mean diameter is 104 millimeters and the standard deviation is 6 millimeters. Find the probability that the diameter of selected bearing is greater than 117 millimeters.

(117 - 104)/6 = -2.17 (according to table = area 0.0150) Correct Answer: 0.0150 CH. 6

Find the area under the normal curve to the right of z = -1.35.

-Look up z=-1.35 which is 0.0885 -Subtract this area from 1. (1 - 0.0885 = .9115) CH. 6

Find the value of z such that 0.383 of the area lies between -z and z.

1 - .383 = .617 / 2 = .3085 ( z is -0.50) the ANSWER is z = 0.50. lose the negative CH. 6.2

Stem and Leaf

1) Arrange data in order 2) Separate the data according to the first digit 3) Display can be made by using the leading digit as the stem and the trailing digit as the leaf CH. 2

528 numbered chips (1-528). What is the probability of drawing the chip numbered 167

1/528=0.0019

President needs to fill 3 remaining spots on cabinet. Has 14 eligible candidates. How many different ways can the members of the cabinet be appointed? Order matters

14P3=2184 CH. 4

How many ways can a doctor visit 4 patients during the morning rounds.

4!=24 CH. 4

What is the probability that a customer is male and lives in "other?" Given info: total customers 1615, total miles that live in other is 67.

67/1615 = .0415 CH. 4

Probability: A box contains 16 green marbles and 19 white marbles. What is the probability of choosing a "white" marble if the first marble (chosen w/o replacement) was a green marble.

A green marble was drawn first, making the total marbles 34. 19/34 = 0.5588 CH. 4

Results sometimes produce flawed conclusions which can be a form of

Bias CH. 1

Select the number of modes and enter the modes, if any. 5, -5, 5, 13, 5, 13, 13 no mode, unimodal, bimodal, and multimodal

Bimodal, Mode 1 = 5, mode 2 = 13. because 5 and 13 are tied for the greatest number of occurrences, the data is bimodal. CH. 3

In how many ways can the letters in the word 'Combination' be arranged?

C=1 O=2 M=1 B=1 N=2 I=2 A=1 T=1 equals 11 11!/1!2!1!1!2!2!1!1! Correct answer: 4,989,600 CH. 4

Graph of Triangle skewed to the right with lines labeled A, B, C, left to right. Find Mean, Median, and Mode.

Mean: The distribution is skewed to the right, so the mean will be the measure of center farthest to the right. Here it is represented by line C Median: The median is the value that divides the area of the distribution in half. Here it is represented by line B. Mode: The mode is always located at the peak of a distribution, so it is line A. CH. 3

Use mean, median or mode when studying this data set: The salary of players on a professional baseball team

Median. Remember: 1. For qualitative data, the mode should be used. 2. For quantitative data the man should be used, unless the data set contains outliers or is skewed. 3. For quantitative data sets that are skewed or contain outliers, the median should be used. CH. 3

Find probability using binomial formula:

P(X=x)=nCx times P^x(1-p)^(n-x) where x= the number of successes, n= the number of trials, and p= the probability of getting a success on any trial Use binomial formula and TI-84. Do the equation individually and sum up the results. "Total Probability" CH. 5

_____ is/are created when there is a problem with either the participation, or lack of participation, of those chosen for the study.

Participation Bias CH. 1

Find the value of z such that 0.04 of the area lies to the right of z.

Scan table A for 0.04 (z-score is -1.75) Drop the negative. The ANSWER is 1.75. CH. 6

Suppose ACT English scores are normally distributed with a "mean of 20.3" and a "standard deviation of 6.1." A university plans to send letters of recognition to students whose scores are in the top "14%." What is the minimum score required for a letter of recognition? Round to nearest tenth, if necessary.

Scan table for top 14%. Scan for .14 (z= -1.08) Make -1.08 +1.08 and Use z-score formula to find X. 1,08=(X-20.3)/6.1 X= 26.9 X represents the minimum ACT English sore required for a letter of recognition is 26.9. CH. 6

Classify distribution: Tail flows to the right

Skewed to the right CH. 2

Find variance using previous table: (square standard deviation)

Standard Deviation: STAT option 1 ENTER Enter Data L1 STAT Calc-option 1: 1-Var Stats L1, L2 ENTER ENTER sideways 6 is S.D. Then you square the S.D. Answer is: 2.4 CH. 5

Find Expected Value: Table: X 4 5 6 7 8 L1 P(X=x) 0.2 0.1 0.2 0.1 0.4 L2

Steps to find Expected Value: -Enter data STAT edit L1 and L2 STAT: Calc 1:1- VAR STATS ENTER 2ND L1, 2ND L2 Answer: 6.4 CH. 5

Using table from above (expected value): P(X< or = 6)

Sum x's that are less than or equal to 6: P(X< or = 6)= P(X=4) + P(X=5) + P(X=6) = 0.2 + 0.1 + 0.2 = 0.5 CH. 5

Identify the sample chosen for the study: The amount of money 15 out of 38 students in your biology class spend on books in a semester

The 15 students in your biology class CH. 1

Complement

The complement of an event is the event not occurring. Thus, the complement of Event A is Event A not occuring. CH. 4

A pair of dice: What is the probability of rolling a sum greater than 5?

There are 36 possible outcomes when rolling two dice. -Remember there are 4 ways to roll a 5 and 36 total outcomes possible. -Compute the probability for each sum greater than 5 on up to the sum of 12. Then ADD up these computations. 5/36+6/36+5/36+4/36+3/36+2/36+1/36=26/36=.7222 CH. 4

Box of 33 red, 73 white, and 85 blue Marbles. Probability of selected marble NOT being red.

There are a total of 191 marbles-of which 158 are not red. 158/191=0.8272 CH. 4

Assume the random variable X has a binomial distribution with the given probability of obtaining a success. P(X>4), n=5, p=0.3 Use Binomial Distribution Function: P(X=x)=nCx times p^x(1-p)^(n-x) by using binompdf (n,p,x) in TI-84

Use binompdf (n,p,x) in TI-84 for P(X=0) thru P(X=4) and sum all the probabilities. P(X=0) = 0.16807 P(X=1) = 0.36015 P(X=2) = 0.3087 P(X=3) = 0.1323 P(X=4) = 0.02835 Sum of all of the above: 0.002 CH. 5

Find the value of z such that 0.11 of the area lies ""to the left"" of z.

Use table to find z - score related to 0.11 which is -1.23 -1.23 is the value of z such that 0.11 of the area lies to the left of z. CH. 6

Find the area under the normal curve ""to the left"" of z = -127

Use table to find z=-127 which area is .1020 Since "to the left of," the area under the curve is .1020 CH. 6

Time-series graph

is a picture of how the data changes over time. Ex: average weekly quiz scores. CH. 2

Cross-sectional graph

is a picture of the data at a given moment in time

What is the value of z divides the standard normal distribution so that half the area is on one side and half is on the other?

ANSWER IS 0 (zero) CH. 6.4

How to determine whether or not the distribution is a probability distribution? X / 4 8 9 P(X=x) / 1/8 1/2 1/8

NO, since the sum of the probabilities is not equal to 1. 1/8+1/2+1/8=3/4, which is less than 1. -Sum of all of the probabilities must equal 1. -Probability of any value must be between 0 and 1, inclusively. CH. 5

New car colors. Suppose 50% of this population prefers the color green. If 12 buyers are randomly selected, what is the probability that exactly a third of the buyers would prefer green?

Probability that 4 out of 12 buyer prefer green: Use binompdf (n,p,x) in TI-84 n=12 p=.5 x=4 binompdf (12, .5, 4) ENTER = .1208496094 or .121 CH. 5

The weight of the steers in a herd are distributed normally. The Standard Deviation is 200 LBS and the mean steer weight is 800 lbs. Find the probability that the weight of a randomly selected steer is between 539 and 1120 lbs.

Probability that the weight of a randomly selected steer is between 539 and 1120: Subtract z-score (using 539) from z-score (using 1120)= answer to the question. [(1120-800)/200] - [(539-800)/200] = 1.6 - (-1.3) = [z-score 1.6] - [z-score -1.31]= From Table: 0.9452 - 0.0951 = 0.8501 CH. 6

How to find lower class boundary

To find class boundary, add the upper limit of one class to the lower limit of the next class and divide by 2. CH. 2

Relative Frequency

To find the relative frequency, divide the frequency of the class by the sample size. Answer in fraction form CH. 2

Find the area under the normal curve to the left of z = -1.87 and to the right of z = 0.95.

To the left of -1.87 (=0.0307) is 0.0307 To the right of 0.95 (=0.8289) is 1 - 0.8289 = 0.1711 0.0307 + 0.1711 = 0.2018 Correct Answer is 0.2018 CH. 6

P(x> or = 15), n=17, p=0.7

Use binompdf (n,p,x) in TI-84 Since it is > or = to 15, add all probabilities X=15, X=16, and X=17. Answer: 0.077 CH. 5

A certain insecticide kills 60% of all insects in a lab experiment. Sample of 6 insects exposed to insecticide. What is the probability that exactly 4 insects will SURVIVE. Use Binomial Distribution Function: P(X=x)=nCx times p^x(1-p)^(n-x) by using binompdf (n,p,x) in TI-84

You are asked to calculate the prob. that exactly 4 insects will survive **THINK: Insects survive or die n= total # of insects-6 p= prob that insects will survive. 100%-60%=40%=0.4 X= number of insects not killed by insecticide-4 input into binompdf (6, 0.4, 4) = 0.138 CH. 5