statistics final

What formula is used: suppose two fair six-sided die are rolled, one blue and one red. Find the probability of rolling a four on the red die, and a two on the blue die.

General Multiplication rule: P( A and B)=P(A)*P(B|A)

What formula is appropriate for this question: If one card is drawn from a standard deck of 52 playing cards, determine the probability of selecting a jack or a spade.

General addition rule: P(A or B)= P(A) + P(B)-P(A and B)

there are 26.1% of five syllable sequences 317 five syllable sequences 61 of that type does the data indicate that the population proportion of this type of 5 syllable squence is different from the text of platos

Ho: p=0.261 Ha: p=/=0.261 Level of significance: 0.1 Test statistic 61/317=0.1924 (0.1924-0.261)/√0.261(1-0.261)/317)=-2.78 -On the z table find the -2.7 on the left and 0.08 at the top. That gives us 0.0027. Since =/=, this is 2 tailed so we need the area to the left and right. 2(0.0027)<0.0054 reject the Ho

Nominal

In name only. Ex: hair color.

Sample Variance

how far each value in the dataset are from the mean. 1. Find the mean (2,5,6,1) 2+5+6+1/4=3.5 2. Subtract the mean from each of the values and square the results (2-3.5)^2 (5-3.5)^2 (6-3.5)^2 (1-3.5)^2= 5.67

Test whether p1>p2 at the 0.05 level of significance. Sample Data: x1=368, n=541, x2=351, n2=593

1. H: p1=p2. Ha:p1>p2 2. Level of significance=0.05 3. 368/541=0.68. 351/593=0.59 (368+351)/(541+593)=0.63 (0.68-0.59)/√0.63(1-0.63) √1/54+1/593)=3.08 Find 3.0 on the left side of the z table and 0.08 at the top=0.9990 1-0.9990=0.001<0.005 -Reject Ho

The equation for a regression line of data is: 3.53x+37.92 What would be the predicted sales if a sales representative traveled 0 miles. Is the answer reasonable?

3.53(0)+37.92=37.92 No, this is not reasonable.

Find the sample mean of the following data set: 65, 66, 67, 66, 67, 71

67 Solve: Add all of the numbers up in the data set. This equals 402. Then, divide that number by the number of values in the data set. 402/6=67

Ho:103 Ha:=/=103. n=35 x=99.9. s=5.9

99.9-103/√5.9/35=-3.108 35-1=34 Find 24 on the t table. There is no 34 so we use 35. One the row 35 find 3.108. Our # falls between 2.996 and 3.340 at the top under two tailed tests. We are between 0.005 and 0.002

Random sampling

each member of the population has an equal chance of being selected

Quantitative variables

variables that can be counted or measured

A brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this machine follows a normal distribution with a mean of 12.18 and a std dev of 0.04. Find the probability that the bottle contains fewer than 12.08 ounces of beer.

0.0062 Solve: 1. subtract the mean from the X (12.08-12.18=-0.1) 2. Divide the answer from step one by the standard deviation. (-0.1/0.04=-2.5) 3. Find that number on the z table (-2.5 on the left and 0.0 at the top=0.0062)

suppose two fair six-sided die are rolled, one blue and one red. Find the probability of rolling a four on the red die, and a two on the blue die.

0.0278 Solve: 1. Find the probability of event one: 1/6 2. Find the probability of event two: 1/6 3. Multiply the probability of the two events: 1/6*1/6=1/36 or 0.0278

Find the area under the standard normal curve between z=1.5 and z=2.5

0.0606 1. Find z=1.5 on the left of the z-table. At the top find 0.0. (0.9332) 2. Find 2.5 on the left side of the z-table. At the top find 0.0. (0.9938) 3. Subtract the largest answer from the two above from the smallest answer from the two above. (0.9938-0.9332)=0.0606

In a recent survey 60% of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 7 of them favor building the health center.

0.157 Solve: 1. Find 14 on the left side of the binomial table. 2. Find 7 under that. 3. Find 0.60 at the top of the table. =0.157

If one card is drawn from a standard deck of 52 playing cards, determine the probability of selecting a ten or a diamond.

0.308 Solve: 1. Find the probability of drawing a ten (4/52) 2. Find the probability of drawing a diamond (13/52) 3. Find the probability of the two events occurring together (There is only 1 ten of diamonds in a deck so 1/52) 4. Plug into the formula: 4/52+13/52-1/52=0.308

Based on data from the NHS, women ages 18-24 have average systolic blood pressure of 114.8 with a standard deviation of 13.1. Systolic bp in women age 18-24 follow normal distribution . If 10 women from the population are randomly selected find the probability that their mean systolic bp is greater than 115.

0.4801 Solve: 1. Divide 13.1 (standard deviation) by the square root of the population (10). 13.1/sqr.root 10=4.14258 2. Subtract the average (114.8) by the probability (115). Divide that by the standard deviation (13.1). Divide that by the square root of the population (10). 3. Solve: 115-114.8/13.1/sqr. root 10=0.048 or 0.05. 4. Find o.o on the left side of the z table and 0.05 at the to =0.05199. 6. 1-0.5199= 0.4801

he following probability distribution tells us the probability that a certain soccer team scores a certain number of goals in a given game: X: 0, 1, 2, 3, 4, P(x): 0.18, 0.34, 0.35, 0.11, 0.02 Calculate the mean and the standard deviation

1. Calculate the mean. Multiply each variable in the table by the probability, then add them together. 0 * 0.18 + 1 * 0.34 + 2 * 0.35 + 3 * 0.11 + 4 * 0.02 = 1.45 goals. 2. Calculate the standard deviation. Subtract the mean from each x value squared. Then multiply it by the probability. (0-1.45)^2*0.18=0.3785 (1-1.45)^2*0.34=0.0689 (2-1.45)^2*0.35=0.1059 (3-1.45)^2*0.11=0.2643 (4-1.45)^2*0.02=0.1301 3. The standard deviation is the square root of the sum of the values. √(.3785 + .0689 + .1059 + .2643 + .1301) = 0.9734 4. The sample variance is the standard deviation squared. 0.9734^2=0.9475

A recent survey found that 70% of adults over 50 wear sunglasses while they drive. In a random sample of ten adults, what is the probability that at least 6 wear sunglasses.

1. Find the sub category 6 under 10 in the binomial table. 2. At the top find 0.70. 3. Add up all of the numbers under 0.70 starting at 6 and ending at 10. =0.8497 or 0.850

Jimmy conducts telephone interviews of 1015 adult residents of NYC about the cost to rent an apartment in the city. Of those sampled, 500 responded that they considered apartment rent to be too high. Construct a 90% confidence interval.

1. Obtain a point estimate. 500/1015=0.49261 2. Determine a 90% confidence interval. find 0.90 at the bottom of the z table. This gives us 1.645. 1.645√0.493(1-0.493)/1015=0.025814 3. Find the limits. LL: 0.4926-0.025814=0.467 UL:0.4926+0.025814=0.518 -We are 90% confident that the population proportion is between 0.467 and 0.518

Based on a poll conducted by the CDC, 862 of 1013 randomly selected adults said that they always wear seatbelts. Construct and interpret a 95% confidence interval for the proportion of adults who always wear their seatbelts.

1. Pointe Estimate p=862/1013=0.85093 2. Margin of Error a. find 0.95 at the bottom of the z table for a critical value of 1.95. b. set up the formula. 1.96 Square root 0.85093(1-0.85093)/1013=0.0219327 3. Limits LL: 0.85093-0.0219327=0.82900 UL: 0.85093+0.0219327=0.872871 We are 95% confident that the proportion is between 0.829 and 0.873

Find the probability of exactly nine girls being born out of 10 births.

1. There is a 50% probability that a baby is born and female. 2. On the left side of the binomial table find 9 in the subcategory under 10. 3. At the top of the table find 0.50 0.0098 or rounded to the third decimal place is 0.010

Jimmy has a bag of marbles, and he has a 25 percent chance of picking a blue marble. Then he draws out one marble and returns it 12 times. You're asked how many times should he get a blue marble.

1. Turn the percent into a decimal. So 25% becomes 0.25 2. Find the number of attempts made. Jimmy drew out a marble 12 times so the number of attempts is 12. 3. multiple the attempts by the percent probability. 0.25*12=3

The children at an elementary school were asked to name their favorite color. Construct a frequency distribution and a relative frequency distribution based on the data: Purple Purple Green Red Blue Blue Blue Purple Blue Green Blue Green Red Red Red Green Blue Red Blue Yellow

1. at the left side of the table the category is listed (in this case color). 2. Beside that is the frequency. (Count the number of each color that occurs) 3. Beside the frequency is the relative frequency (the frequency divided by N. Calculate the N by adding up all of the frequencies) Purple: 3, 0.15 green: 4, 0.20 red: 5, 0.25 blue: 7, 0.35 yellow: 1, 0.05

A study was recently done on drinking and driving. 400 accidents occurred on a Saturday night and were analyzed. Determine the probability of an accident involving alcohol given there were 3 or more vehicles involved.

1. divide the number of accidents in the 3 or more category where alcohol was involved by the total under the 3 or more category. 20/50 or 0.4

A school is including a mile run in its fitness test. The time for boys has a mean of 450 seconds and a standard deviation of 50. Find the probability that a randomly selected boy can run the mile in less than 335 seconds.

1. subtract the mean from the x. (335-450=-115) *make sure to do the equation one at a time in the calculator. Not all at once* 2. Divide the result by the standard deviation. (115/50=-2.3) 3. Find -2.3 on the left side of the z-table and 0.0 at the top. (That gives us 0.0107) *if we were finding something greater then you would subtract the answer from 1*

A school is including a mile run in its fitness test. The time for boys has a mean of 460 seconds and a standard deviation of 60. Between what times do we expect 95% of boys to run the mile?

1. turn the percentage into a decimal. 95%=0.95. 2. Subtract the percent from one. 1-0.95=0.05 2. Divide by two. 0.05/2=0.025. 3. In the body of the z table, find 0.025 or the closest number to it. The closest number we could find is 0.0250. To the left of that is -1.9. At the top is 0.06. 4. Solve the equation with a negative -1.96 and with a positive 1.96 so get the two times. (-1.96)60+460=342.4 (1.96)60*460=577.6 so 95% of the boys run the mile between 342.4-577.6

Based on data from the NHS, women ages 18-24 have average systolic blood pressure of 114.8, with a standard deviation of 13.1. SBP in women age 18-24 follow normal distribution. If 10 women from the population are randomly selected, find the probability that their mean systolic BP is greater than 115.

13.1/√10=4.14258 *Do this formula one piece at a time* 115-114.8/13.1/√10=0.048 or 0.05 -Find 0.0 on the left side of the z table and 0.05 at the top. That gives us 0.5199 1-0.5199=0.4801

Find the sample standard deviation for the data set: 65, 66, 67, 66, 67, 71

2.098 Solve: 1. Find the mean of the data set 2. Subtract the mean from each of the data values and square them (Ex: (65-57)^2 ). Write down the answers 3. Add up the answers from step 2 4. Subtract the number of values from one. (Ex: if there are six values in the data set then 6-1=5) 5. Divide the value from step 3 by the value from step 4. 6. Get the square root of the answer from step 5.

Describe the shape of the distribution

Bell shaped/symmetrical

Find the formula that suits the question: In a healthy handwashing study conducted by the Bradley Corporation, it was found that roughly 65% of Americans operate the toilet flusher in public restrooms with their foot. Suppose a random sample of 10 adults is selected, find the probability that at least 8 of them flush the toilet with their foot.

Binomial table.

What formula is appropriate for the question: A quiz consists of 10 true or false questions. To pass the quiz a students must get at least 8 questions correct. If the student guesses on each question, what is the probability of passing?

Binomial table.

Interval

Ex: degrees in celsius or fahrenheit or the year.

Ratio

Ex: weight (0 is meaningless).

Median

Middle number

A simple random sample of size N is drawn from a population that is normally distributed and does not have outliers. The sample mean, x, is 50, and the sample standard deviation, s, is 8. Construct a 98% confidence interval for m if the sample size, n, is 9.

Point Estimate x=50 Error Find 8 on the left side of the T table and 0.98 at the top. That gives us 2.896 2.8968*8/√9=7.72267 50+7.72267=57.72 50-7.72267=42.28

Continuous variable

Quantitative variable that can be measured. Ex: weight or blood pressure.

What.formula is used to find the sample mean of a data set? (Ex: 2,3,5,4,6)

Sample mean: x= Exi/n

Describe the shape of the distribution

Skewed left

Describe the shape of the distribution

Skewed right

What formula is appropriate to answer the question: A brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this machine follows a normal distribution with a mean of 12.18 and a std dev of 0.04. Find the probability that the bottle contains fewer than 12.08 ounces of beer.

Standard Normal z table Z Score: z=x-u/std dev

What formula is appropriate for the question: Find the area under the standard normal curve to the left of z=-2.15

Standard normal z table

Find the appropriate formula for the question: the avg scores for all golfers for a particular course has a mean of 75 and a std dev of 3.5. suppose 49 golfers played the course today. Find the probability that the avg score for all the golfers exceeded 76.

Standard normal z table Z score for a sample mean: z= x-u/std dev/sqr root of n

You are dealt one card from a standard 52 deck. Find the probability of not drawing a 5.

There are 4 five cards in a standard deck. There are two ways to complete this formula. 1. 1-4/52=0.923076 or 2. 52-4=48 48/52=0.923076

A traffic officer is compiling information about the hour of the day relating to speeding tickets. He comes up with a correlation coefficient of 0.52. What does this mean?

There is a moderate positive linear correlation.

A random sample of 51 fatal car crashes in 2017 in which the driver had a positive blood alcohol concentration (BAC) from the National Highway Traffic Safety Administration results in a mean BAC of 0.167 g per decilliter. With a std dev of 0.010 g/dl A. A histogram of blood alcohol concentration in fatal accidents shows that BAC are highly skewed right. Explain why a large sample is needed to construct a confidence interval for the mean BAC of fatal crashed with a positive BAC. B. Determine and interpret a 90% confidence interval for the mean BAC in fatal crashed in which the driver had positive BAC.

a. S=0.010, X=0.167, N=51, C=0.90 N>30- the central limit theorem applies b. Point Estimate: x=0.167 Error: -On the T -Table find 50 on the left side and .90 at the top=1.676. 1.676 (0.010/sqr.root. 51)= 0.002347 0.167+ or - 0.002347= UL: 1.6934. LL: 0.164

mean

average (add up all the numbers in a data set, then divide by how many there are)

Convenience sampling

chosen based soley on convenience

Multistage Sampling

combine sampling methods.

Mode

most frequently occurring score

Ordinal

natural order. Ex: shirt sizes smallest to largest.

Qualitative Variables

neither discrete or continuous. Ex: hair color or shirt size

Parameter

numerical summary of a population

Discrete variable

quantitative variables that can be counted. Ex: number of heart attacks, number of hospital visits.

Classify the variable: The cholesterol levels of a group of people the day after thanksgiving.

quantitative, continuous, ratio.

Cluster Sampling

select all individuals in a randomly selected collection group. Ex: Select people who all live on the same floor of a building ad then group them together in groups based on a different characteristic.

Systematic Sampling

select every nth or kth individual.

stratafied sampling

separate populations into non overlapping groups obtain a simple random sample from each group.

35 math majors, 29 music majors, and 26 history majors are all randomly selected from 585 math majors, 279 music majors, and 393 history majors. What sampling technique is used?

stratified

range

the difference between the highest and lowest scores in a distribution. (find the smallest number in the data set and subtract it from the largest number)

Study of Statistics

the science of collecting, organizing, analyzing, and interpreting data in order to make decisions

One card is drawn from a standard deck of 52 cards what is the probability of drawing a 2 or a club.

there are 4 two cards in a deck, 13 club cards in a deck and 1 two of clubs in a deck. 4/52+13/52-1/52=4/13 or 0.30769