MATH 210 Homework 3.1 and 3.2
A defunct website listed the "average" annual income for Florida as $35,031. What is the role of the term average in statistics? Should another term be used in place of average?
The term average is not used in statistics. The term mean should be used for the result obtained by adding all of the sample values and dividing by the total number of sample values.
Listed below are measured amounts of caffeine (mg per 12oz of drink) obtained in one can from each of 14 brands. Find the range, variance, and standard deviation for the given sample data. Include appropriate units in the results. Are the statistics representative of the population of all cans of the same 14 brands consumed? 36, 36, 40, 33, 0, 56, 59, 57, 50, 30, 35, 40, 0, 0 a) The range of the sample data is ___ (brands, brands^2, mg per 12oz of drink, mg per 12oz of drink^2) b) The standard deviation of the sample data is ___ (brands, brands^2, mg per 12oz of drink, mg per 12oz of drink^2) c) The variance of the sample data is ___ (brands, brands^2, mg per 12oz of drink, mg per 12oz of drink^2) d) Are the statistics representative of the population of all cans of the same 14 brands consumed? -The statistics are likely representative of the population of all cans of these brands that are consumed because the results from any sample of the 14 brands will be typical of the population. -The statistics are not necessarily representative of the population of all cans of these brands that are consumed because it is necessary to have at least 5 of each brand in order to get a sample that is representative of the population. -The statistics are likely representative of the population of all cans of these brands that are consumed because each brand is being represented in the sample. -The statistics are not necessarily representative of the population of all cans of these brands that are consumed because each brand is weighted equally in the calculations. It is unlikely that each of the 14 brands of soda are consumed equally.
a) 59 mg per 12oz of drink b) 20.5 mg per 12oz of drink c) 420 mg per 12oz of drink^2 d) The statistics are not necessarily representative of the population of all cans of these brands that are consumed because each brand is weighted equally in the calculations. It is unlikely that each of the 14 brands of soda are consumed equally.
Use the body temperatures, in degrees Fahrenheit, listed in the accompanying table. The range of the data is 3.2°F. Use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the actual standard deviation of the data rounded to two decimal places, 0.71°F, assuming the goal is to approximate the standard deviation within 0.2°F. 98.1, 98.1, 97.5, 98.4, 99.1, 98.7, 98.3, 98.4, 98.1, 98.8, 99, 99, 98.6, 96.6, 97.2, 99.1, 97.5, 97.3, 99.1, 98.1, 98.3, 98.6, 98, 96.9, 98.8, 96.9, 98.8, 98.8, 99.7, 97.9, 97.5, 98.1, 97.7, 99.7, 99.1, 99.1, 98.2, 98, 98.5, 98.7, 96.7, 98.3, 98.7, 98.3, 98.2, 98, 97.9, 98.2, 98.2, 98.5, 97.5, 98.9, 96.5, 97.7, 97.8, 96.7, 97.1, 97.5, 97.7, 98.5, 97.3, 98.2, 97.8, 97.4, 98.6, 98.9, 99, 99.1, 97.5, 98, 96.8, 97.8, 99.6, 98.1, 98.6, 98.3, 98.2, 98.5, 98.3, 99.1, 99, 98.7, 98.7, 97.7, 99, 97.9, 98.7, 98.4, 98.7, 98.5, 98.3, 97.2, 97.6, 99.2, 98.3, 97.5, 98.3, 99, 98, 98.3, 98.9, 97.7, 97.9, 97.5, 97.6, 96.9 a) The estimated standard deviation is ___ °F. Compare the result to the actual standard deviation. b) The estimated standard deviation is ___ (more than 0.2° greater than, more than 0.2° less than, within 0.2° of) the actual standard deviation. Thus, the estimated standard deviation (does not meet, meets) the goals.
a) .8 b) within 0.2° of, meets
Use the F-scale measurements of tornadoes listed in the accompanying table. The range of the data is 5.0. Use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the actual standard deviation of the data, 1.3, assuming a difference of 0.3 or greater would mean a significant difference. 1, 3, 5, 3, 1, 4, 0, 2, 4, 3, 1, 2, 3, 1, 1, 3, 1, 0, 1, 5, 2, 0, 0, 1, 0, 0, 3, 0, 2, 1, 2, 0, 0, 0, 2, 0, 2, 1, 3, 2, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 2, 5, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 3, 2, 2, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 2, 0 a) Using the range rule of thumb, the standard deviation is approximately ___ Compare the result to the actual standard deviation. b) The estimated standard deviation is ___ (more than 0.3 greater than, more than 0.3 less than, within 0.3 of) the actual standard deviation. Thus, the estimated standard deviation ___ (is not, is) significantly different from the actual standard deviation.
a) 1.3 b) within 0.3 of, is not
A sample of blood pressure measurements is taken from a data set and those values (mm Hg) are listed below. The values are matched so that subjects each have systolic and diastolic measurements. Find the mean and median for each of the two samples and then compare the two sets of results. Are the measures of center the best statistics to use with these data? What else might be better? Systolic Diastolic 154 63 111 71 135 66 160 86 145 56 100 88 130 51 156 84 117 74 143 59 a) The mean for systolic is ___mm Hg and the mean for diastolic is ___mm Hg b) The median for systolic is ___mm Hg and the median for diastolic is ___mm Hg. c) Compare the results. Choose the correct answer below. -The mean and median appear to be roughly the same for both types of blood pressure. -The median is lower for the diastolic pressure, but the mean is lower for the systolic pressure. -The mean is lower for the diastolic pressure, but the median is lower for the systolic pressure. -The mean and the median for the systolic pressure are both lower than the mean and the median for the diastolic pressure. -The mean and the median for the diastolic pressure are both lower than the mean and the median for the systolic pressure. d) Are the measures of center the best statistics to use with these data? -Since the systolic and diastolic blood pressures measure different characteristics, only measures of center should be used to compare the data sets. -Since the sample sizes are equal, measures of center are a valid way to compare the data sets. -Since the sample sizes are large, measures of center would not be a valid way to compare the data sets. -Since the systolic and diastolic blood pressures measure different characteristics, a comparison of the measures of center doesn't make sense. e) What else might be better? -Because the data are matched, it would make more sense to investigate whether there is an association or correlation between the two blood pressures. -Since measures of center would not be appropriate, it would make more sense to talk about the minimum and maximum values for each data set. -Because the data are matched, it would make more sense to investigate any outliers that do not fit the pattern of the other observations. -Since measures of center are appropriate, there would not be any better statistic to use in comparing the data sets.
a) 135.1, 69.8 b) 139, 68.5 c) The mean and the median for the diastolic pressure are both lower than the mean and the median for the systolic pressure. d) Since the systolic and diastolic blood pressures measure different characteristics, a comparison of the measures of center doesn't make sense. e) Because the data are matched, it would make more sense to investigate whether there is an association or correlation between the two blood pressures.
The brain volumes (cm3) of 50 brains vary from a low of 902 cm3 to a high of 1488 cm3. Use the range rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 161.3 cm3, assuming the estimate is accurate if it is within 15 cm3. a) The estimated standard deviation is ___cm3. b)Compare the result to the exact standard deviation. -The approximation is not accurate because the error of the range rule of thumb's approximation is greater than 15 cm3. -The approximation is accurate because the error of the range rule of thumb's approximation is greater than 15 cm3. -The approximation is not accurate because the error of the range rule of thumb's approximation is less than 15 cm3. -The approximation is accurate because the error of the range rule of thumb's approximation is less than 15 cm3.
a) 146.5 ((max-min)/4) ((1488-902)-4) b) The approximation is accurate because the error of the range rule of thumb's approximation is less than 15 cm3.
Listed below are the numbers of hurricanes that occurred in each year in a certain region. The data are listed in order by year. Find the range, variance, and standard deviation for the given sample data. Include appropriate units in the results. What important feature of the data is not revealed by any of the measures of variation? 4, 8, 15, 10, 4, 2, 20, 11, 1, 2, 6, 7, 6, 2 a) The range of the sample data is ___ (years, years^2, hurricanes, hurricanes^2) b) The standard deviation of the sample data is ___ (years, years^2, hurricanes, hurricanes^2) c) The variance of the sample data is ___ (years, years^2, hurricanes, hurricanes^2) d) What important feature of the data is not revealed through the different measures of variation? -The measures of variation reveal no information about the scale of the data. -The measures of variation reveal nothing about how the numbers of hurricanes are spread. -The measures of variation reveal nothing about the pattern over time. -The measures of variation do not reveal the difference between the largest number of hurricanes and the smallest number of hurricanes in the data.
a) 19 hurricanes b) 5.5 hurricanes c) 30 hurricanes^2 d) The measures of variation reveal nothing about the pattern over time.
Find the (a) mean, (b) median, (c) mode, and (d) midrange for the given sample data. An experiment was conducted to determine whether a deficiency of carbon dioxide in the soil affects the phenotype of peas. Listed below are the phenotype codes where 1=smooth-yellow, 2=smooth-green, 3=wrinkled-yellow, and 4=wrinkled-green. 1, 4, 1, 3 ,1, 3, 2, 4, 3, 3, 4, 3, 3, 4 e) Do the results make sense? -Only the mean, median, and mode make sense since the data is numerical. -All the measures of center make sense since the data is numerical. -Only the mean, median, and midrange make sense since the data is nominal. -Only the mode makes sense since the data is nominal.
a) 2.8 b) 3 c) 3 d) 2.5 e) Only the mode makes sense since the data is nominal.
Find the (a) mean, (b) median, (c) mode, and (d) midrange for the data and then (e) answer the given question. Listed below are the jersey numbers of 11 players randomly selected from the roster of a championship sports team. What do the results tell us? 9, 14, 19, 66, 75, 73, 49, 35, 56, 15, 97 e) What do the results tell us? -The jersey numbers are nominal data and they do not measure or count anything, so the resulting statistics are meaningless. -Since only 11 of the jersey numbers were in the sample, the statistics cannot give any meaningful results. -The mean and median give two different interpretations of the average (or typical) jersey number, while the midrange shows the spread of possible jersey numbers. -The midrange gives the average (or typical) jersey number, while the mean and median give two different interpretations of the spread of possible jersey numbers.
a) 46.2 b) 49 c) There is no mode d) 53 e) The jersey numbers are nominal data and they do not measure or count anything, so the resulting statistics are meaningless.
A certain group of test subjects had pulse rates with a mean of 70.5 beats per minute and a standard deviation of 12.2 beats per minute. Use the range rule of thumb to identify the limits separating values that are significantly low or significantly high. Is a pulse rate of 74.9 beats per minute significantly low or significantly high? a) Significantly low values are ___ beats per minute or lower. b) Significantly high values are ___ beats per minute or higher. c) Is a pulse rate of 74.9 beats per minute significantly low or significantly high? -Significantly low, because it is more than two standard deviations below the mean. -Significantly high, because it is more than two standard deviations above the mean. -Neither, because it is within two standard deviations of the mean. -It is impossible to determine with the information given.
a) 56.1 ((mean)- 2x( standard deviation)) b) 94.9 ((mean)+ 2x( standard deviation)) c) Neither, because it is within two standard deviations of the mean.
Listed below are pulse rates (beats per minute) from samples of adult males and females. Find the mean and median for each of the two samples and then compare the two sets of results. Does there appear to be a difference? Male Female 53 85 73 91 59 95 86 91 69 62 97 76 57 87 95 89 58 88 55 69 58 78 90 95 58 86 91 74 82 84 a) The mean for males is ___beats per minute and the mean for females is ___ beats per minute. b) The median for males is ___ beats per minute and the median for females is ___ beats per minute. c) Compare the results. Choose the correct answer below. -The mean and the median for females are both lower than the mean and the median for males. -The mean and the median for males are both lower than the mean and the median for females. -The median is lower for males, but the mean is lower for females. -The mean is lower for males, but the median is lower for females. -The mean and median appear to be roughly the same for both genders. d) Does there appear to be a difference? -The pulse rates for males appear to be higher than the pulse rates for females. -There does not appear to be any difference. -Since the sample size is small, no meaningful information can be gained from analyzing the data. -The pulse rates for females appear to be higher than the pulse rates for males.
a) 72.1, 83.3 b) 69, 86 c) The mean and the median for males are both lower than the mean and the median for females. d) The pulse rates for females appear to be higher than the pulse rates for males.
Using the accompanying table of data, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.5. (All units are 1000 cells/μL.) Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? 321, 298, 285, 262, 216, 249, 388, 218, 506, 351, 329, 189, 243, 249, 185, 391, 195, 234, 229, 214, 221, 168, 335, 241, 275, 254, 576, 157, 185, 246, 396, 218, 239, 197, 202, 204, 226, 219, 183, 253, 316, 214, 243, 170, 311, 338, 246, 214, 240, 226, 372, 234, 227, 310, 310, 229, 373, 186, 242, 289, 229, 179, 261, 331, 185, 184, 294, 229, 288, 240, 217, 262, 205, 184, 245, 308, 253, 234, 207, 265, 225, 214, 254, 169, 359, 166, 323, 169, 295, 269, 164, 169, 376, 218, 282, 338, 243, 223, 254, 262, 230, 388, 241, 236, 232, 270, 261, 246, 274, 279, 331, 246, 327, 299, 204, 197, 175, 329, 208, 227, 263, 257, 282, 232, 222, 257, 220, 183, 324, 263, 211, 233, 251, 207, 238, 255, 305, 275, 393, 182, 304, 232, 190, 236, 131, 245, 293 a) Using Chebyshev's theorem, what is known about the percentage of women with platelet counts that are within 2 standard deviations of the mean? At least ___% of women have platelet counts within 2 standard deviations of the mean. What are the minimum and maximum possible platelet counts that are within 2 standard deviations of the mean? b) The minimum possible platelet count within 2 standard deviations of the mean is ___. The maximum possible platelet count within 2 standard deviations of the mean is ___.
a) 75 b) 124.1, 386.1
Below are the jersey numbers of 11 players randomly selected from a football team. Find the range, variance, and standard deviation for the given sample data. What do the results tell us? 23, 48, 8, 61, 65, 86, 59, 52, 57, 68, 22 a) Range= ___ b) Sample standard deviation= ___ c) Sample variance= ___ d) What do the results tell us? -Jersey numbers on a football team vary much more than expected. -The sample standard deviation is too large in comparison to the range. -Jersey numbers are nominal data that are just replacements for names, so the resulting statistics are meaningless. -Jersey numbers on a football team do not vary as much as expected.
a) 78 b) 23.2 c) 538.1 d) Jersey numbers are nominal data that are just replacements for names, so the resulting statistics are meaningless.
The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 245.9 and a standard deviation of 60.3. (All units are 1000 cells/μL.) Using the empirical rule, find each approximate percentage below. a) What is the approximate percentage of women with platelet counts within 2 standard deviations of the mean, or between 125.3 and 366.5? b) What is the approximate percentage of women with platelet counts between 185.6 and 306.2? a) Approximately ___% of women in this group have platelet counts within 2 standard deviation of themean, or between 125.3 and 366.5. b) Approximately ___% of women in this group have platelet counts between 185.6 and 306.2.
a) 95 b) 68
