Ch 2
what is the purpose of a density curve
to model a distribution with a continuous function so you can take any interval and the area under the curve will equal the area of the bars in a histogram
you can roughly locate the median of a density cure by eye because it is
the point that divides the area under the curve into two unequal parts
empirical rule
the property of normal distributions. it is the same for all normal distributions regardless of the values of the mean and standard deviation
if tim scored a 72 on a test and his calculated z score was -1.32, what does that mean
tim scored 1.32 times the standard deviation below the mean
z scores round to
2 decimal places
one method to determine if the given distribution that you suspect is about normal
close to satisfying the empirical rule, first we need a distribution that "looks" about normal. use graphing calculator to asses
what happens to a distribution if each data value is transformed linearly by multiplying or dividing each data value by a number?
it will change the measures of position and variability in the same way, but the shape will not change
steps when solving a problem where you want to find normal proportions
1. think of the problem in terms of the observed variable, call that variable x. in terms of x you want to find the proportion of data where x is greater than the other #, sketch the normal distribution and shade the region under the curve where x is greater 2. standardize the x variable by converting it to a z score then sketch a standard normal distribution and shade the area under the curve to the right of the calculated z score 3. find the area sketched in step 2 by looking up the area in the table 4. write a conclusion in the context of the problem (proportion, proportion, percentage, expected #, expected #)
if the total area of the bars in a histogram = 100%, what is the value of the area under the density curve? how would it be expressed as a proportion?
100%, 1
suppose that scores on a certain IQ test are normally distributed with mean 110 and standard deviation 15. then about 40% of the scores are between
102 and 118
a company produces packets of soap powder that are labeled "giant size 32 ounces." the actual weight of soap powder in a box has a normal distribution with a mean of 33 oz and a standard deviation of .7 oz 95% of packets actually contain more than x oz. of sap powder. what is x?
31.85
entomologist heinz kaefer has a colony of bongo spiders in his lab. there are 1000 adult spiders in the colony, and their weights are normally distributed with mean 11 grams and standard deviation 2 grams. about how many spiders are there in a colony which weigh more than 12 grams?
310
the graduate record examination (GRE) are widely used to help predict the performance of applicants to graduate schools. the range of possible scores on the GRE is 200 to 900. the psychology department at a university finds that the scores of its applicants on the quantitative GRE are approximately normal with mean of 544 and standard deviation of 103. find the proportion of GRE scores are below 500?
33.36%
percentiles round to
4 decimal places
IQs among undergraduates at mountain tech are approximately normally distributed. the mean undergraduate IQ is 110. about 95% of undergraduates have IQs between 100 and 120. the standard deviation of these IQs is about...
5
the graduate record examination (GRE) are widely used to help predict the performance of applicants to graduate schools. the range of possible scores on the GRE is 200 to 900. the psychology department at a university finds that the scores of its applicants on the quantitative GRE are approximately normal with mean of 544 and standard deviation of 103. find the proportion of applicants whos scores are between 328 and 853
97.58%
density curve
a curve that is always on or above the horizontal axis and has an area of exactly 1. it describes the overall pattern of a distribution. the area under the curve and above any interval of values on the horizontal axis is proportional of all observations that fall in that interval
standard normal distribution
a distribution where you convert all the values in a normal distribution to z scores then the distribution of these z scores will also be normal with a mean of 0 and standard deviation of 1. this is represented by n(0,1)
relative cumulative frequency (ogive)
a graph that can be made with percentiles
what happens to a dot plot of new data if you increase it by a %
all measures will increase by that %
normal curves
density curves that are unimodal (one peak) and symmetric
a taxi driver charged an initial fee of $2.50 plus $2 per mile. he calculated his mean distance and standard deviation at the end of the month ( mean = 6.5 miles, standard deviation = 2 miles). he wants to know what the average and standard deviation was for the fares that month in dollars. how do you calculate that? what happens to the shape?
fare = 2.5 + 2 (each mile) mean fare = 2.5 + 2(6.5) = $15.50 standard deviation = 2 x 2 = $4 the shape will not change
how to find relative frequency
frequency in each interval / the total # of values (frequencies) in the data
what does being in the 95th percentile mean?
if you are in the 95th percentile, that means you scored better than 95% of those who took that test. In other words, only 5% scored equal to or better than you.
a sample was taken of the salaries of 20 employees of a large company. suppose each employee in the company receives a $3,000 raise for next year (each employee's salary is increased by $3,000). the mean salary for the employees will... the median salary for the employees working for the company will... the standard deviation of the salaries for the employees will... the interquartile range of the salaries for the employees will... the z - scores of the salaries for the employees will...
increase by $3,000 increase by $3,000 be unchnaged be unchanged be unchanged
if katie scored a 93 on a test and her calculated z score was 2.14, what does that mean
katie scored 2.14 times the standard deviation above the mean
what is the use of a method for accessing normality
since the normal distribution is so useful it is so important to know when you can consider a distribution "normal enough" to call it an approximately normal distribution
how to find cumulative relative frequency
sum up all the relative frequency values up to and including that value. the last value in the column must be 100% and theses percentages are in fact the percentiles for the data within the class
what do u and o mark
u marks the exact center of the distribution and o marks the distance from u where the curve changes from concave down to concave up. the curves extend infinitely along the horizontal axis and are asymptotic to the asix
how do you describe the relative position of an individual
with a percentile
the mean number of days that the midge chaoborus spends in its larval stage is 14.1 days, with a standard deviation of 2.2 days. this distribution is skewed toward higher values. what is the z score for an individual midge that spends 12.7 days in its larval stage?
-0.64
using the standard normal distribution tables, the area under the standard normal curve corresponding to Z> -1.22 is
.8888
what happens to a distribution if each data is transformed linearly by adding or subtracting some value to each data value?
adding or subtracting the same value to each number in a data set will change measures of position (min, q1, med, q3, max, and mean) in the same way. but measure of variability (iqr, range, and standard deviation) and shape will not change
how can you describe performance relative to the whole class?
by calculating the percentile
normal curve is a
theoretical mathematical model for data distributions that are unimodal and have a symmetrical "bell" shape
in a normal distribution... (6 points)
- 68, 95, 99.7 rule -mean and median are at the center and about the same -q1 and q3 are about the same distance away from the mean -shape is symmetric, single peaked, bell shaped -standard deviation measures the variability (width) of a normal distribution -if a normal distribution plot is nearly linear then the distribution is about normal
- 3 minutes 45 seconds - 20% of the songs - 3 minutes - skewed right
- approximately what is the median rolling stones song length? - approximately what percent of rolling stones songs are over 5 minutes in length? - approximately what is the 20th percentile of rolling stones song lengths? - what is the shape of this distribution?
- z score = .42 the student that had a self concept score of 62 scored .42 times the standard deviation above the mean - skewed left
- calculate and interpret the z score for the student that had a self concept score of 62 - on the graph above, make a rough sketch of a density curve for these data. how would you describe the shape of this density curve?
birth weights at a local hospital have a normal distribution with a mean of 110 oz. and a standard deviation of 15 oz. the proportion of infants with birth weights between 125 oz and 140 oz is about
.136
birth weights at a local hospital have a normal distribution with a mean of 110 oz. and a standard deviation of 15 oz. the proportion of infants with birth weights under 95 oz is about
.159
items produced by a manufacturing process are supposed to weigh 90 grams. the manufacturing process is such, however, that there is variability in the items produced and they do not all weigh exactly 90 grams. the distribution of weights can be approximated by a normal distribution with a mean of 90 grams and a standard deviation of 1 gram. about what percentage of items will either weigh less than 87 grams or more than 93 grams?
.3%
using the standard normal distribution tables, the area under the standard normal curve corresponding to -0.5 < Z < 1.2 is
.5764
the graduate record examination (GRE) are widely used to help predict the performance of applicants to graduate schools. the range of possible scores on the GRE is 200 to 900. the psychology department at a university finds that the scores of its applicants on the quantitative GRE are approximately normal with mean of 544 and standard deviation of 103. what proportion of GRE scores are above 800?
.6387%
using the standard normal distribution tables, the area under the standard normal curve corresponding to Z < 1.1 is
.8643
the scores on a university examination are normally distributed with a mean of 62 and a standard deviation of 11. if the bottom 5% of students will fail the course, what is the lowest mark that a student can have and still be awarded a passing grade?
45
a soft drink machine can be regulated so that it discharges an average mean oz per cup. if the ounces of fill are normally distributed with a standard deviation of .4 oz, what value should mean be set at so that 98% of 6 oz cups will not overflow?
5.18
the graduate record examination (GRE) are widely used to help predict the performance of applicants to graduate schools. the range of possible scores on the GRE is 200 to 900. the psychology department at a university finds that the scores of its applicants on the quantitative GRE are approximately normal with mean of 544 and standard deviation of 103. calculate and interpret the 34th percentile of the distribution of applicants' GRE scores
501. 34% of those that take the GRE score below 502
the 5 number summary of the distribution of 316 scores on a statistics exam is: 0, 226, 31, 36, 50. the scores are approximately normal. the standard deviation of test scores must be about
7.5
here is a list of exam scores for mr. williams calculus class: 60 61 61 65 72 75 75 78 81 81 85 89 91 98. what is the percentile of the person whose score was 85?
71%
standard normal probabilities table
a method for finding the area to a certain side of a z score graph
the normal probability plot
a more specialized graphical display that can help you decide whether a normal density curve is appropriate for a set of data
(start of 2.2) density curves
a smooth curve describing the overall pattern/shape of the data in the graph (histogram, stem plot, or dot plot) of a large number of observations when it is so regular
rules for the empirical rule
about 68% of the observations fall within 1 standard deviation of the mean, about 95% of the observations fall within 2 standard deviations of the mean, and about 99.7%/100% of the observations fall within 3 standard deviations of the mean
when will a normal probability plot appear linear and what does that indicate?
any unusual deviations from a linear normal probability plot may appear linear and it may indicate that the distribution is not normal. it is necessary to understand how to use this plot, particularly for small data sets in which normality cannot be determined by a histogram or the empirical rule
how do we typically refer to distributions as
approximately normal since no distribution is exactly normal
why cant you explain why you think a distribution is normal saying that q1 and q3 are about the same distance from the mean
because it only suggests symmetry and not all symmetric ones are normal. it would be better to plot the points into the calculator, see if the probability plot is linear, and if it is then it is normal
in the case of jenny's test scores even though 86 on the statistics test was higher than the 82 on the english test, her z scores for these were .99 and 1.5 respectively, so she actually performed .... relative to her classes on the english test
better
the five number summary of the distribution of scores on the final exam in Psych 001 last semester was 18, 39, 62, 76, 100. the 80th percentile was
between 76 and 100
another more time consuming method to check for normality of a distribution that only works for large data sets is to
check and see how closely the distribution of your data satisfies the empirical rule. this can be done by finding the mean and the standard deviation, and determing what percent of the data that fall within one, two, and three standard deviations of the mean. if those results closely follow the empirical rule it is reasonable to assume the distribution is normally distributed
what do standardized scores (z scores) allow us to do
compare scores from two different distributions on a common scale
mathematical models
density curves that model the idealized descriptions of the overall pattern in the data
normal distribution
described by a normal density curve, any particular normal distribution is completely specified by 2 numbers, its mean and standard deviation. the mean is at the center and the standard deviation is distance from mean to the point where the curve changes from concave down to concave up (infection points) on either side of the mean
how to construct an ogive graph
draw axes where the horizontal axis is the range of the classes and the vertical axis is the cumulative relative frequency percentages. plot points coordinating to the cumulative relative frequency in each class interval at the left endpoint of the next class interval. connect these points with a smooth curve
how do you calculate the percentile? (use an example of a students score compared to her class)
her rank ( low to high ) / total # of students or her rank ( high to low ) / total # of students
a standard score describes
how far a particular score is above or below the mean
jack and jill are both enthusiastic players of a certain computer game. over the past year, jack's mean score when playing the game is 12,400 with a standard deviation of 1500. during the same period, jill's mean score is 14,200 with a standard deviation of 2000. they devise a fair contest: each one will play the game once, and they will compare z scores. jack gets a score of 14,000 and jill gets a score of 16,000. who won the contest, and what were each of their z scores
jacks z = 1.07; jills z = 1.11; jill wins the contest
what should you always, always, always do?
look at a graph (histogram, dot plot, stem and leaf plot) of the distribution before assessing normality. the distribution should be unimodal and roughly symmetric before we even start looking at the normal probability plot. and as always, be careful about outliers
where do you mark mean and median on a density curve
mean is at the balance point, median is at the 50th percent tile
what are you doing when performing a linear transformation?
standardizing the data by calculating the z scores. to do this you enter the original data in the calculator, then transform each value by subtracting the mean then dividing by the standard deviation
what is a drawback with the normal probability plot?
that all the data must be in a single list, there is no option for a count or frequency list like you have for a histogram or box plot. it would therefore be difficult to enter all the class data from 1 d into a list and graph the normal probability plot. in that case you best bet is to make a histogram to assess normality
- all of the above - 20% - 72%
the density curve shown below in figure 1-1 is a horizontal line at 0.2 from 0 to 5. - for this density curve, which of the following is true? (options are it is symmetric, the total area under the curve is 1, the median is 2.5, and the mean is 2.5) - for this density curve, what percent of the observations are greater than 4? - for this density curve, what percent of the observations lie between .2 and 3.8?
how to we refer to normal distribution
the greek letter "u" to represent mean and the greek letter "o" to represent standard deviation
how to read a standard normal probabilities table
the left column is the value of the z score given to the nearest tenth. the column headings at the top of the table are values for the z score to the nearest hundredth place. the body of the table provides the value of the area to the left of the corresponding z score
if you standardize every test score from Mr. Bowman's class what would the new mean and standard deviation be and how would the shape be effected?
the mean = 0, standard deviation = 1, and the shape does not change
mean = 100; standard deviation = 65
the normal curve below describes the death rates per 100,00 people in developed countries in the 1990's. the mean and standard deviation of this distribution are approximately
you can roughly locate the mean of a density curve by because it is
the point at which the curve would balance if made of solid material
z score
the result of standardizing, represents the number of standard deviations the data value is above of below the mean of that data
skewed right
when a basketball player makes a pass to a teammate who then scores, he earns an assist. below is a normal probability plot for the number of assists earned by all players in the national basketball association during the 2010 regular season. what shape is the distribution?
20 students were asked to guess the age of a man in a photograph. here are their guesses: 44, 43, 48, 37, 44, 40, 33, 42, 43, 41, 50, 49, 43, 46, 46, 45, 43, 38, 39, 41. are these guesses approximately normally distributed? proved evidence to support your answer
yes because when i entered the points into my calculator and graphed it as a probability plot, it was approximately linear
can we fully describe the density curve for a normal distribution in terms of just u and o
yes since all normal distributions will have the same bell shape
can density curves occur in other shapes?
yes, commonly being skewed left, skewed right, and symmetric. a density curve can be symmetric and not normal
how do you determine the percent of scores in the data table that fall within one standard deviation of the mean?
you first find range : mean plus standard deviation and then mean minus standard deviation. you count the amount of scores the fall in between those two numbers then divide it out of the total number of scores. the convert to a percent
(2.1 quiz starts) you are told that your score on an exam is at the 85 percentile of the distribution of scores. this means that
your score was higher than approximately 85% of the people who took this exam
formula for calculating a z score
z = x - mean / standard deviation
