MATH-164 - Chapter 3
identify the given statement as either true or false. The standard deviation can be negative.
False
Violent crimes include rape, robbery, assault, and homicide. The following is a summary of the violent-crime rate (violent crimes per 100,000 population) for all states of a country in a certain year. Q1 = 272.8, Q2 = 387.9, Q3 = 528.3 (a) Interpret these results.(b) Determine and interpret the interquartile range.(c) The violent-crime rate in a certain state of the country in that year was 1 comma 459. Would this be an outlier?(d) Do you believe that the distribution of violent-crime rates is skewed or symmetric?
(a) Interpret these results.Q1: 25% of the states have a violent-crime rate that is 272.8 crimes per 100,000 population or less.Q2: 50% of the states have a violent-crime rate that is 387.9 crimes per 100,000 population or less.Q3: 75% of the states have a violent-crime rate that is 528.3 crimes per 100,000 population or less. (b) Determine and interpret the interquartile range. The interquartile range is 528.3 - 272.8 = *255.5 crimes per 100,000 population. The middle 50% of all observations have a range of 255.5 crimes per 100,000 population.* Hints: The interpretation of the interquartile range is similar to that of the range and standard deviation. That is, the more spread a set of data has, the higher the interquartile range will be. (c) The violent-crime rate in a certain state of the country in that year was 1459. Would this be an outlier? The lower fence is 272.8 - 1.5 x 255.5 = -110.45 crimes per 100,000 population. The upper fence is 528.3 + 1.5 x 255.5 = 911.55 crimes per 100,000 population. Since violent-crime rate in a certain state of the country in that year is greater than the upper fence (1459>911.55), it is an outlier. (d) Do you believe that the distribution of violent-crime rates is skewed or symmetric?The difference between Q 1 and Q 2 is quite a bit less than the difference between Q 2 and Q 3 (387.9-272.8>528.3-387.9). In addition, the outlier in the right tail of the distribution implies that the distribution is skewed right. Thus, the distribution of violent-crime rates is skewed right.
A graph is an ogive of a standardized test's scores. The vertical axis in an ogive is the cumulative relative frequency and can also be interpreted as a percentile. (a). Find and interpret the percentile rank of a test score with a value of 120. (the respectively vertical axis value is 50%)(b) Find and interpret the percentile rank of a test score with a value of 140. (the respectively vertical axis value is 90%)
(a). Find and interpret the percentile rank of a test score with a value of 120.A test score of 120 corresponds to the 50th percentile rank since this percentage of test scores are less than or equal to a test score with a value of 120. Hint: Look at the graph and match the number given (120 on x-axis) by the y-axis to get the percentile. (b) Find and interpret the percentile rank of a test score with a value of 140.A test score of 140 corresponds to the 90th percentile rank since this percentage of test scores are less than or equal to a test score with a value of 140.
Suppose the first class in a frequency table of quantitative data is 0-4 and the second class is 5-9. What is the class midpoint of the first class?
2.5 add the two lowest values from the frequency tables then divide by 2. 0+5= 5/2=2.5
Violent crimes include rape, robbery, assault, and homicide. The following is a summary of the violent-crime rate (violent crimes per 100,000 population) for all states of a country in a certain year. Complete parts (a) through (d). Q1=271.8, Q2=387.9, Q3=529.1
25% of the states have a violent-crime rate that is 271.8 crimes per 100,000 population or less. 50% of the states have a violent-crime rate that is 387.9 crimes per 100,000 population or less. 75% of the states have a violent-crime rate that is 529.1 crimes per 100,000 population or less.
Complete the following statement about the Empirical Rule. According to the Empirical Rule, if a distribution is bell-shaped, then approximately _______of the data will lie within 1 standard deviation of the mean; approximately ______ of the data will lie within 2 standard deviations of the mean; approximately _____ of the data will lie within 3 standard deviations of the mean.
68% 95% 99.7%
sample size
A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equal chance of occurring. The sample is then called a simple random sample.
Explain the meaning of the accompanying percentiles. (a) The 10th percentile of the head circumference of males 3 to 5 months of age in a certain city is 41.0 cm. (b) The 80th percentile of the waist circumference of females 2 years of age in a certain city is 49.8 cm. (c) Anthropometry involves the measurement of the human body. One goal of these measurements is to assess how body measurements may be changing over time. The following table represents the standing height of males aged 20 years or older for various age groups in a certain city in 2015. Based on the percentile measurements of the different age groups, what might you conclude?
A. 10% of 3- to 5-month-old males have a head circumference that is 41.0 cm or less. B. 80% of 2-year-old females have a waist circumference that is 49.8 cm or less. C. At each percentile, the heights generally decrease as the age increases. Assuming that an adult male does not grow after age 20, the percentiles imply that adults born in 1990 are generally taller than adults who were born in 1950.
What can be said about a set of data with a standard deviation of 0?
All the observations are the same value.
Which of the accompanying boxplots likely has the data with the larger standard deviation? Why?
Boxplot II likely has the data with the larger standard deviation because the boxplot appears to have a greater spread, which likely results in a larger standard deviation.
When an observation that is much larger than the rest of the data is added to a data set, the value of the median will increase substantially.
FALSE- Because median is least affected by outliers
Identify the given statement as either true or false. The standard deviation is a RESISTANT MEASURE of spread.
False
Which of the following are resistant measures of dispersion?
Interquartile Range
the arithmetic mean of a variable
Is computed by adding all the values of the variable in the data set and dividing by the number of observations.
Which of the following are resistant measures of central tendency?
Median
Range formula
Range = X(max) - X(min)
The mean finish time for a yearly amateur auto race was 186.51 minutes with a standard deviation of 0.378 minute. The winning car, driven by Roger, finished in 185.53 minutes. The previous year's race had a mean finishing time of 112.8 with a standard deviation of 0.125 minute. The winning car that year, driven by Tammy, finished in 112.56 minutes. Find their respective z-scores. Who had the more convincing victory?
Roger: -2.59Tammy: -1.92 Roger had a more convincing victory because of a lower z-score
Outliers
Sample values that lie very far away from the vast majority of the other sample values
Greek letters
The letters of the Greek alphabet most commonly used for parameters are μ (the population mean) and σ (pronounced 'sigma').
Which measures are used in the five-number summary?
The five-number summary of a data set consists of the smallest data value, the first quartile Q1, the median, the third quartile Q3, and the largest data value.
z-score
a measure of how many standard deviations you are away from the norm (average or mean)
Explain the meaning of the following percentiles in parts (a) and (b). (a) The 10th percentile of the weight of males 36 months of age in a certain city is 13.0 kg. (b) The 95th percentile of the length of newborn females in a certain city is 53.3 cm.
a. 10% of 36-month-old males weigh 13.0 kg or less, and 90% of 36-month-old males weigh more than 3.0 kg. b. 95% of newborn females have a length of 53.3 cm or less, and 5% of newborn females have a length that is more than 53.3 cm.
The table shows the weekly income of 20 randomly selected full-time students. If the student did not work, a zero was entered. (a) Check the data set for outliers. (b) Draw a histogram of the data. (c) Provide an explanation for any outliers.
a. The outlier(s) is/are 3096. (usually the biggest numbers that stand out from the data) c. A student with unusually high income A student providing false information Data entry error
The sample arithmetic mean,
computed using sample data (statistic) x(x-bar"), is a statistic that is computed using data from individuals in a samp
resistant
if observations that are extreme (very large or small) relative to the data do not affect its value substantially. So the median is resistant, but the mean is not resistant.
The population standard deviation of a variable
is the square root of the sum of squared deviations about the population mean divided by the number of observations in the population, N
For a distribution that is symmetric, which of the following is true?
mean EQUALS median
For a distribution that is skewed right, which of the following is true?
mean GREATER THAN median
multimodal
more than two modes
Parameter
numerical summary of a population ,
The sample standard deviation, s,
of a variable is the square root of the sum of squared deviations about the sample mean divided by n−1, where n is the sample size. The formula is given as
For a distribution that is skewed right, the median is left of center_________.
of the box.
Suppose the data represent the inches of rainfall in April for a certain city over the course of 20 years. Given the quartiles Q1=1.910, Q2=3.535, and Q3=5.355, determine the lower and upper fences. Are there any outliers, according to this criterion? Are there any outliers in the given data set? What are the outliers? Select the correct choice below and fill in any answer boxes in your choice.
q3-q1=IQR LF q1-1.5(IQR) UF q3+1.5(IQR) IQR=5.355-1.910=3.445 LF 1.910 -1.5(3.445) LF 1.910 -5.1675 = -3.2575 or -3.258 The lower fence is negative -3.258 UF 5.355+1.5(3.445) UF 5.355+5.1675 = 10.5225 or 10.523 The upper fence is 10.523. No, all the values are between the lower and upper fences. (if the data set is below the LF (-3.258) or above the UL (10.523) then it has outliers. There are no outliers.
Interquartile Range (IQR)
the difference between the first and third quartiles is the range of the middle 50% of the observations in a data set. That is, the IQR is the difference between the first and third quartiles and is found using this formula IQR=Q3−Q1.
The range R
the difference between the largest and the smallest data value
Median
the middle score in a distribution; half the scores are above it and half are below it. The median of a variable is the value that lies in the middle of the data when arranged in ascending order. We use M to represent the median.
Mode
the most frequently occurring score(s) in a distribution. The mode of a variable is the observation of the variable that occurs most frequently in the data set. -To compute the mode, tally the number of observations that occur for each data value. -The data value that occurs most often is the mode. -If no observation occurs more than once, we say that the data have no mode. -A set of data can have no mode, one mode, or more than one mode.
For a distribution that is skewed left, the left whisker is longer than
the right whisker.
For a distribution that is skewed left, the left whisker is longer than __________.
the right whisker.
For a distribution that is symmetric, the left whisker is the same length as ___________.
the right whisker.
Bimodal
two modes
The _______ represents the number of standard deviations an observation is from the mean.
z-score
z score formula
z=(x-mean)/standard deviation z-score=x−μ/σ= If a data value is larger than the mean, the z-score is positive. If a data value is smaller than the mean, the z-score is negative. If the data value equals the mean, the z-score is zero. A z-score measures the number of standard deviations an observation is above or below the mean. For example, a z-score of 1.24 means the data value is 1.24 standard deviations above the mean. A z-score of −2.31 means the data value is 2.31 standard deviations below the mean.
The sum of the deviations about the mean always equals
zero
The population arithmetic mean
μ (pronounced "mew"), is a parameter that is computed using data from all the individuals in a population.
Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 16. Use the empirical rule to determine the following. (a) What percentage of people has an IQ score between 68 and 132? (b) What percentage of people has an IQ score less than 68 or greater than 132? (c) What percentage of people has an IQ score greater than 148?
(a) 95% (Type an integer or a decimal.) (b) 5% (Type an integer or a decimal.) (c) .15% (Type an integer or a decimal.)
Male and female populations of tortoises under 80 years old are represented by age in the table below. Complete parts (a) through (d). Age 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 Males 10 17 17 15 22 23 17 12 Females 6 8 15 20 24 22 17 14
(a) Approximate the population mean and standard deviation of age for males. μ=41.42 years σ=20.85 years (b) Approximate the population mean and standard deviation of age for females. μ=44.5 years σ=19.19 years (c) Which gender has the higher mean age? Females have the higher mean age. (d) Which gender has the higher dispersion in age? Males have the greater dispersion. (for the population mean use stat crunch, stat, summary stat, group/bin, bin in (age), counts in (males), click (consecutive lower limits), click (mean), click compute) (for the standard deviation use stat crunch, stat, summary stat, group/bin, bin in (age), counts in (males), click (limits), click (unadj. std.dev), click compute) Do the same for the females.
Violent crimes include rape, robbery, assault, and homicide. The following is a summary of the violent-crime rate (violent crimes per 100,000 population) for all states of a country in a certain year. Q1=273.8, Q2=387.4, Q3=528.3
25% of the states have a violent-crime rate that is 273.8 crimes per 100,000 population or less. 50% of the states have a violent-crime rate that is 387.4 crimes per 100,000 population or less. 75% of the states have a violent-crime rate that is 528.3 crimes per 100,000 population or less.
True or False: When comparing two populations, the larger the standard deviation, the more dispersion the distribution has, provided that the variable of interest from the two populations has the same unit of measure.
True, because the standard deviation describes how far, on average, each observation is from the typical value. A larger standard deviation means that observations are more distant from the typical value, and therefore, more dispersed.
