stats final exam

Consider the following data. 15,14,3,1,−5,−1015,14,3,1,−5,−10 Copy Data Step 3 of 3 : Determine if the data set is unimodal, bimodal, multimodal, or has no mode. Identify the mode(s), if any exist.

-If all of the data values occur only once, or they each occur an equal number of times, the data set is considered to have no mode. -If only one value occurs the most, then the data set is said to be unimodal. -If exactly two values occur equally often and more than all the others, then the data set is said to be bimodal. -If more than two values occur equally often and more than all the others, then the data set is multimodal. >ANS: NO MODE.

Consider the following data: >: x6,7,8,9,10,P(X=x)P(X=x)0.1,0.1,0.2,0.2,0.4 Copy Data

-In calculator: press stat and go to Edit,choose edit:1 and enter, input first values in L1, press enter after every value, same for second values entered in L2. -press quit, the stat and go to Calc menu. choose 1:1 var stats and press enter. press L1,(2ND,1) and L2 and enter. > p(x<7)=.1, stand dev. = 1.34, ex=8.7

Performing a Hypothesis Test

-State the null and alternative hypotheses. -Determine which distribution to use for the test statistic, and state the level of significance. -Gather data and calculate the necessary sample statistics. -Draw a conclusion and interpret the decision.

Properties of the Standard Normal Distribution

-The standard normal distribution is bell-shaped and symmetric about its mean. -The standard normal distribution is completely defined by its mean, μ=0μ=0, and standard deviation, σ=1σ=1. -The total area under the standard normal normal distribution curve equals 11. -The x-axis is a horizontal asymptote for the standard normal distribution curve.

2.1 prob determining lower class boundary: class | frequency 6-12 3 13-19 14 20-26 4 27-33 5 34-40 8

-find upper number before 4th class, 26. -determine lower number of the 4th class, 27. >26+27= 53 53/2= 26.5 ANS: 26.5

determining upper limit of the same data set from previous.

-find upper number of first class, 12 -find lower number of second class, 13 >12+13=25 25/2=12.5

Determine the cumulative frequency for the fourth class.

-first, second, third & fourth class summed together.

Determine the frequency of each class in the table shoterm-32wn.

-how many numbers are between the values listed on the class.

determing class width of the same data set from previous.

-lower limit of first class, 20 -lower limit of second class,29 >29-20=9

determine class width.

-subtract lower limit of first class from lower limit of second class.

Calculate the weighted average balance for the three-month period. Month | avrg. monthly value october 2251.33 november 2490.51 december 1478.27

-the data weight is the monthly values. -there are 31,30, 31 days in each month. >(31)(2251.33)+(30)(2490.51)+(31)(1478.27)/31+30+31 =190332.90/92=2068.84

Consider the following data. 15,14,3,1,−5,−1015,14,3,1,−5,−10 Copy Data Step 2 of 3 : Determine the median of the given data.

-which value is in the middle, 2.

The weighted mean is calculated as follows: >hand >calculator

(83(0.4)+98(0.2)+90(0.1)+87(0.3))/(0.4+0.2+0.1+0.3) (¯x)=87.9 ​Therefore, Walter's final grade for the class is 87.9. >go to STAT > EDIT. Enter the x values (Walter's grades) into L1 and the corresponding weights in decimal form into L2. Next, go to STAT > CALC and choose 1-Var Stats. This time after pressing Enter we will add L1, L2 so we are computing 1-Var Stats L1, L2.​ >enter top values in L1, bottom values in L2.

Determine the relative frequency for the third class as a simplified fraction. class | frequency 7-13 13 14-20 14 21-27 12 28-34 7 35-41 4

- add up all frequency values, 13+14+12+7+4=50 -find frequency for 3rd class, 12 >freq. of 3rd class/ sample size: 12/50

Properties of a Normal Distribution

-A normal distribution is bell-shaped and symmetric about its mean. -A normal distribution is completely defined by its mean μ and standard deviation σ. -The total area under a normal distribution curve equals 11. -The x-axis is a horizontal asymptote for a normal distribution curve.

Using a Normal Distribution to Approximate a Binomial Distribution

-Determine the values of n and p. Verify that the conditions np≥10n⁢p≥10 and n(1−p)≥10n⁢1−p≥10 are met. -Calculate the values of the mean and standard deviation of the binomial random variable using the formulas μ=npμ=n⁢p and σ=np(1−p)⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√σ=n⁢p⁢1−p. -Use a continuity correction to determine the interval corresponding to the given value of x. -Draw a normal curve using the mean and standard deviation calculated in Step 3, and label it with the information given in the problem. -Using either a z-value with normal distribution tables or available technology, find the appropriate area under the normal curve.

Determine the cumulative frequency for the second class.

-First and second class are summed together, 9+14= 23 -Answer, cummulative freq. = 23.

Using a TI-83/84 Plus Calculator to Find Areas under the Standard Normal Curve

-For area to the left of z0z0, P(z≤z0)P⁡z≤z0: Enter normalcdf(−1E99, z0) -For area to the right of z0z0, P(z≥z0)P⁡z≥z0: Enter normalcdf(z0, 1E99) -For area between z1z1 and z2z2, P(z1≤z≤z2)P⁡z1≤z≤z2: Enter normalcdf(z1, z2) -For area to the left of z1z1 plus area to the right of z2z2, P(z≤z1 or z≥z2)P⁡z≤z1 or z≥z2: Enter 1−normalcdf(z1, z2)

Using the Cumulative Normal Distribution Tables to Find Areas under the Standard Normal Curve

-For area to the left of z0z0, P(z≤z0)P⁡z≤z0: Look up z0z0 -For area to the right of z0z0, P(z≥z0)P⁡z≥z0: Look up −z0−z0 -For area between z1z1 and z2z2, P(z1≤z≤z2)P⁡z1≤z≤z2: Look up z1z1 and z2z2 and subtract area to the left of z1z1 from area to the left of z2z2 -For area to the left of z1z1 plus area to the right of z2z2, P(z≤z1 or z≥z2)P⁡z≤z1 or z≥z2: Look up z1z1 and −z2−z2 and add areas

The director of research and development is testing a new medicine. She wants to know if there is evidence at the 0.01 level that the medicine relieves pain in more than 373 seconds. After performing a hypothesis test, she decides to reject the null hypothesis.

>There is sufficient evidence at the 0.01 level of significance that the medicine relieves pain in more than 373 seconds

A lumber company is making boards that are 2711 millimeters tall. If the boards are too long they must be trimmed, and if they are too short they cannot be used. A sample of 7 boards is made, and it is found that they have a mean of 2706.8 millimeters with a variance of 121. Is there evidence at the 0.025 level that the boards are too short and unusable?

>below. H0: μ=2711 Ha: μ<2711

continuous or discrete? > temp. in Fahrenheit of cities in south Carolina > number of elliptical machines in every YMCA in your state

>continuous >discrete

Trucks in a delivery fleet travel a mean of 90 miles per day with a standard deviation of 19 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives less than 103 miles in a day. Round your answer to four decimal places.

>in a day. μ=90, σ=19 Given informationP(X<103) >103-90/19=.68

Finding the Median of a Data Set

>list data from small to largest, cross out. >STAT > EDIT and entering the data into L1. Next, go to STAT > CALC and choose 1-Var Stats. Press Enter to obtain a list of descriptive statistics. by hand>​1, 4, 5, 7, 8, 8, 9, 9 ​(7+8)/2=7.5

A company has given you the task to research the salaries of a class with 3030 college students and the instructor. Would you be more interested in looking at the mean, median, or mode?

>median

determining frequency of class

Make a tally mark for each data value in the appropriate class. Count the marks to find the total frequency for each class.

Reject the null hypothesis, H0H0, if:

z≤−zαz≤−zα for a left-tailed test z≥zαz≥zα for a right-tailed test |z|≥zα2/|z|≥zα2 for a two-tailed test > conclusions: If p-value ≤α≤α, then reject the null hypothesis. If p-value >α>α, then fail to reject the null hypothesis.

Suppose that salaries for recent graduates of one university have a mean of $⁢25,500 with a standard deviation of $1050$⁢1050. Using Chebyshev's Theorem, what is the minimum percentage of recent graduates who have salaries between $⁢22,350 and $⁢28,650? Round your answer to one decimal place.

> First, 22350-25500/1050=-3,150/1050=-3 & 28650-25500/1050=3150/1050=3 >According to Chebyshev's Theorem, at least 88.9% of data values lie within 33 standard deviations of the mean.

The type and number of fish caught in the Charleston Harbor in March was recorded for a month. The results are recorded in the table below. What is the probability that the next fish caught is a drum or a bluefish?

> add the number of drum and bluefish and / by number of fish

pop or sample? >All college students at UV of Jackson > 359 college students surveyed

> population >sample

Calculate the range, population variance, and population standard deviation for the following data set. If necessary, round to one more decimal place than the largest number of decimal places given in the data. 14,11,6,16,7,11,16,5,13,15

> range, largest&smallest #. 16-5 >pop variance,

Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.61 and a standard deviation of 0.39. Using the empirical rule, what percentage of the students have grade point averages that are between 1.83 and 3.39? >According to the empirical rule, what percentage of data values lie within 2 standard deviations of the mean

>1.83-2.61/0.39=-2 & 3.39-2.61/0.39=2 >95%

You are going to play mini golf. A ball machine that contains 20 green golf balls, 19 red golf balls, 17 blue golf balls, and 22 yellow golf balls, randomly gives you your ball. What is the probability that you end up with a blue golf ball? Express your answer as a simplified fraction or a decimal rounded to four decimal places.

>17/78 >number of blue/ whole number of golfballs.

A soft drink machine outputs a mean of 26 ounces per cup. The machine's output is normally distributed with a standard deviation of 4 ounces. What is the probability of filling a cup between 28 and 31 ounces? Round your answer to four decimal places.

>28 and 31 ounces. μ=26μ=26, σ=4σ=4Given information(28<X<31)P(28<X<31) >Graphical representation of the problemP(28−264< (x−μσ) <31−264)=P(0.5<z<1.25) >p(0.5<z<1.25

Calculate the standard score of the given x value, x=91.6x=91.6, where μ=99.1μ=99.1, σ=3.7σ=3.7. Round your answer to two decimal places.

>91.6-99.1/3.7=-2.02

calculating sample mean >To find the sample mean on a TI-83/84 Plus calculator

>Add the hours together and then divide by 7, which is the number of students in the sample. >go to STAT > EDIT. Enter the values into L1 as shown on the top screenshot below. (Make sure any old data has been deleted first.) Next, press STAT > CALC and choose 1-Var Stats. Press Enter

A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 425 gram setting. It is believed that the machine is underfilling the bags. A 23 bag sample had a mean of 417 grams with a standard deviation of 28. A level of significance of 0.01 will be used. Assume the population distribution is approximately normal. there sufficient evidence to support the claim that the bags are underfilled?

>Answer:H0: μ=425 answer:Ha: μ<425 >statistic. t=x-μ/s/√n >417-425/28/sqrt 23 --1.370

A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 425 gram setting. It is believed that the machine is underfilling the bags. A 23 bag sample had a mean of 417 grams with a standard deviation of 28. A level of significance of 0.01 will be used. Assume the population distribution is approximately normal. Is there sufficient evidence to support the claim that the bags are underfilled?

>Press STAT. Select TESTS. Select option 2:T-Test. Press ENTER. Input the corresponding values. You have the option to input Data or Stats. -Data - If you have the raw data and it is a small enough sample where it is reasonable to enter, you will get a more exact answer using this than the summary statistics. See Adding Info to Lists for further help on entering your data. Stats - Use this option if you were provided summary statistics.

You were asked to solve the following problem.In a recent study at a local college using a simple random sample, 38 students admitted that they had tried alcohol at least once while under the legal drinking age, while 28 students said they had not. Is there enough data to conduct a hypothesis test to see if the percentage of students at the college who admit to drinking underage is the same as the national percentage of 63%?

>n=66, p=.63 >The experiment consists of n=38+28=66

u, x,

>population mean >sample mean

A person rolls a standard six-sided die 5 times. In how many ways can he get 2 fours, 2 sixes, and 1 three?

>use factorial in the calculator. >press math, select PRB, choose 4:!,and enter. Math, PRB, 5!=120 & 2x2x1!=4 , 120/4=30

Finding Probability for Any Normal Curve Tables:

Using the appropriate lower and upper bounds as well as the mean and standard deviation of the distribution, use the normalcdf function on the calculator to compute the area.

The following is a graph of two normal distributions plotted on the same x-axis. 10 and 5 plotted on the same area.

The two distributions have means that differ by 5 units and equal standard deviations.

find class boundary

The upper limit of class one is 54,999. The lower limit of class two is 55,000. Thus, the class boundary between the first two classes is calculated as follows. (54,999+55,000)/2=54999.5

Find the area under the standard normal curve between z=−1.76 and z=0.07. Round your answer to four decimal places, if necessary.

Type "=NORM.S.DIST(z, cumulative)". (Choose TRUE for cumulative for cdf, FALSE is pdf.) Press ENTER.

Find the value of z such that 0.09 of the area lies to the left of z. Round your answer to two decimal places.

Type "=NORM.S.INV(probability)". Press ENTER.

Finding the Value of a Normally Distributed Random Variable for a Given Probability

Using the appropriate area, mean and standard deviation, compute the z-score by using the invNorm function on the calculator.

Eco-Cook juicer has a mean time before failure of 39 months with a standard deviation of 3 months, and the failure times are normally distributed. What should be the warranty period, in months, so that the manufacturer will not have more than 10% of the juicers returned? Round your answer down to the nearest whole number.

We know that the normal distribution tables give the area to the left of a specified z-value. Scanning the interior of the tables for an area of 0.1, we find that the closest corresponding z-value is z=−1.28. Then, we use the appropriate formula to find the value of x. x=z⋅σ+μ≈(−1.28)(3)+39=35.16

Find the total of the areas under the standard normal curve to the left of z1=−1.645 and to the right of z2=1.645. Round your answer to four decimal places, if necessary.

We know that the standard normal distribution tables only give the area to the left of a specified z-value. However, since z1 and z2 are negatives of each other, then, by symmetry, the area to the right of z2 is equal to the area to the left of z1. Thus, to determine the total area in two tails, we should find the area to the left of z1 and double it.Looking up the respective value in the tables, we determine that the area to the left of z1=−1.645 is 0.05. Therefore, the total of the areas to the left of z1z1 and to the right of z2z2 is 2⋅0.05=0.10

sample

a subset of the population

Select the phrase that best completes the following statement. A measure of central tendency is

a typical value in a data set.

Consider the following data. 15,14,3,1,−5,−1015,14,3,1,−5,−10 Copy Data Step 1 of 3 : Determine the mean of the given data.

add all values, divide by # of values.

sample mean is the

arithmetic mean of a set of sample data

quantitative

consists of counts/ measurements (quantities)

An Australian Shepherd breeder has had three litters of puppies from the same set of parents. The following table shows the results from the three litters. In the next litter, what is the probability of a puppy having spots?

count all the puppys with spots and / by all puppys w/without spots.

data

counts, measurements, in observations

continuous data

data that can take on any given value. in any interval, measurements.

frequency distribution

display of values used & how often used.

class boundary

is the value that lies halfway between the upper limit of one class and the lower limit of the next class

qualitative data

labels/ descriptions (qualities)

find upper limit

largest number in the set

class midpoint is the value in the middle of the class, and is given by:

lower limit + upper limit / 2.

Suppose each value in a data set had a constant value of c subtracted from it. How would this affect the mean, median, and mode?

mean, median, mode & range decreased by c.

For the following type of data set, would you be more interested in looking at the mean, median, or mode? State your reasoning. The salary of actors on TV

median, salaries of actors are quantitative data, with outliers.

Suppose babies born in a large hospital have a mean weight of 3316 grams, and a standard deviation of 324 grams. If 83 babies are sampled at random from the hospital, what is the probability that the mean weight of the sample babies would be less than 3267 grams? Round your answer to four decimal places.

normal with μx‾=μ=3316μx‾=μ=3316. σ=Standard Deviation of the population=324 n=Sample Size=83 > σx‾=σ2n⎯⎯⎯⎯⎯⎯√=σn⎯⎯=324/√83 ≈35.563620

frequencies (F)

number of data values in categories of frequency distributions.

parameter

numerical descriptions

ordered array

ordered lsit of data from large to small

population

particular group of interest

discrete data

qualitative data that take sonly particular values, counts.

Find the variance of the following data. Round your answer to one decimal place. x3,4,5,6,7,8,P(X=x)P(X=x)0.2,0.1,0.1,0.2,0.2,0.2 Copy Data

same step as ^. look for variance.

find lower limit

smallest number in the set

Find class width

subtract the lowest # from the highest # in a set. Divide the difference by the number of classes.

population mean

the arithmetic mean of all the values in a population,

Reject the null hypothesis, H0, if:

t≤−tαt≤−tα for a left-tailed test t≥tαt≥tα for a right-tailed test |t|≥tα2/|t|≥tα2 for a two-tailed test

Sample Characteristics

used to describe a sample

variable

value/ characteristic that changes