Question

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 (type integers or decimals rounded to one decimal place as needed.) compare the results. choose the correct answer below.
systolic: 114, 96, 146, 109, 121, 139, 97, 137, 117, 118
diastolic: 57, 74, 59, 56, 67, 58, 91, 69, 54, 76
compare options: a. the mean and the median for the systolic pressure are both lower than the mean and the median for the diastolic pressure. b. the median is lower for the diastolic pressure, but the mean is lower for the systolic pressure. c. the mean and the median for the diastolic pressure are both lower than the mean and the median for the systolic pressure. d. the mean is lower for the diastolic pressure, but the median is lower for the systolic pressure. e. the mean and median appear to be roughly the same for both types of blood pressure.
are the measures of center the best statistics to use with these data?
options: a. since the systolic and diastolic blood pressures measure different characteristics, a comparison of the measures of center doesnt make sense. b. since the sample sizes are large, measures of center would not be a valid way to compare the data sets. c. since the systolic and diastolic blood pressures measure different characteristics, only measures of center should be used to compare the data sets. d. since the sample sizes are equal, measures of center are a valid way to compare the data sets.
what else might be better?
options: a. 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. b. because the data are matched, it would make more sense to investigate any outliers that do not fit the pattern of the other observations. c. because the data are matched, it would make more sense to investigate whether there is an association or correlation between the two blood pressures. d. since measures of center are appropriate, there would not be any better statistic to use in comparing the data sets.

Answer:

First, let's find the mean and median for systolic and diastolic blood pressures.

Systolic Blood Pressure Data: 114, 96, 146, 109, 121, 139, 97, 137, 117, 118

1. Mean Calculation:

• Sum of systolic values: \(114 + 96 + 146 + 109 + 121 + 139 + 97 + 137 + 117 + 118\)
• \(114+96 = 210\); \(210+146 = 356\); \(356+109 = 465\); \(465+121 = 586\); \(586+139 = 725\); \(725+97 = 822\); \(822+137 = 959\); \(959+117 = 1076\); \(1076+118 = 1194\)
• Number of data points (\(n\)) = 10
• Mean (\(\bar{x}\)) = \(\frac{1194}{10} = 119.4\)

1. Median Calculation:

• First, sort the data: 96, 97, 109, 114, 117, 118, 121, 137, 139, 146
• Since \(n = 10\) (even), median is the average of the 5th and 6th values.
• 5th value = 117, 6th value = 118
• Median = \(\frac{117 + 118}{2} = \frac{235}{2} = 117.5\)

Diastolic Blood Pressure Data: 57, 74, 59, 56, 67, 58, 91, 69, 54, 76

1. Mean Calculation:

• Sum of diastolic values: \(57 + 74 + 59 + 56 + 67 + 58 + 91 + 69 + 54 + 76\)
• \(57+74 = 131\); \(131+59 = 190\); \(190+56 = 246\); \(246+67 = 313\); \(313+58 = 371\); \(371+91 = 462\); \(462+69 = 531\); \(531+54 = 585\); \(585+76 = 661\)
• Number of data points (\(n\)) = 10
• Mean (\(\bar{y}\)) = \(\frac{661}{10} = 66.1\)

1. Median Calculation:

• First, sort the data: 54, 56, 57, 58, 59, 67, 69, 74, 76, 91
• Since \(n = 10\) (even), median is the average of the 5th and 6th values.
• 5th value = 59, 6th value = 67
• Median = \(\frac{59 + 67}{2} = \frac{126}{2} = 63\)

Now, let's analyze the options for the first question (Compare the results):

• Systolic mean = 119.4, Systolic median = 117.5
• Diastolic mean = 66.1, Diastolic median = 63
• We can see that the mean and median for systolic are both higher than the mean and median for diastolic. Also, the mean (119.4) and median (117.5) for systolic are close, and the mean (66.1) and median (63) for diastolic are also close. So the mean and median appear to be roughly the same for both types of blood pressure. So the correct option is C.

For the second question (Are the measures of center the best statistics...):

• The data is matched (each subject has both systolic and diastolic measurements). Measures of center (mean, median) describe the central tendency, but since the data is paired, we can investigate the association (like correlation) between systolic and diastolic pressures. Also, checking for outliers or spread (like range, standard deviation) could be better, but among the options:
• Option A: Wrong, because comparing measures of center can make sense (we just did it).
• Option B: Wrong, sample size being large doesn't make measures of center invalid.
• Option C: Correct, because systolic and diastolic are different characteristics (different variables) for the same subjects, so measures of center alone might not be the best; we can investigate the relationship between them.
• Option D: Wrong, equal sample sizes don't make measures of center the only valid way.

So the correct option is C.

For the third question (What else might be better?):

• Option A: Talking about min and max (range) is a measure of spread, but not necessarily better than investigating association.
• Option B: Checking outliers is good, but since data is matched, investigating association is better.
• Option C: Correct, since data is matched (paired), investigating the association (correlation) between systolic and diastolic pressures makes sense.
• Option D: Wrong, because even if measures of center are appropriate, investigating association can give more insights.

So the correct option is C.

Final Answers:

1. For the first (compare results) question: \(\boldsymbol{C}\)
2. For the second (best statistics) question: \(\boldsymbol{C}\)
3. For the third (what else better) question: \(\boldsymbol{C}\)