Question

a study shows that 28% of the population has high blood pressure. the study also shows that 86% of those who do not have high blood pressure exercise at least 90 minutes per week, while 32% of those with high blood pressure exercise at least 90 minutes per week. which of the following relative frequency tables could the study provide? four tables with rows: has high blood pressure, does not have high blood pressure, total; columns: exercise ≥ 90 minutes per week, exercise < 90 minutes per week, total, with various percentages in each cell

Explanation:

Step1: Analyze given percentages

We know:
- Total with high blood pressure (HBP): \(28\%\) (so without HBP: \(100 - 28=72\%\))
- Of those without HBP, \(86\%\) exercise \(\geq90\) mins/week. So percentage of population without HBP and exercise \(\geq90\): \(0.72\times0.86 = 0.6192\approx62\%\)
- Of those with HBP, \(32\%\) exercise \(\geq90\) mins/week. So percentage of population with HBP and exercise \(\geq90\): \(0.28\times0.32 = 0.0896\approx9\%\) (Wait, no—wait, the "of those with HBP" exercise \(\geq90\) is \(32\%\), so the percentage in the table for "Has HBP" and "Exercise \(\geq90\)" should be \(28\%\times32\%?\) No, wait, the table is relative frequency. Let's re - express:

Let's define:
- Let \(P(HBP)=0.28\), \(P( eg HBP)=0.72\)
- \(P(\text{Exercise}\geq90| eg HBP) = 0.86\), so \(P(\text{Exercise}\geq90\cap eg HBP)=P( eg HBP)\times P(\text{Exercise}\geq90| eg HBP)=0.72\times0.86 = 0.6192\approx62\%\)
- \(P(\text{Exercise}\geq90|HBP)=0.32\), so \(P(\text{Exercise}\geq90\cap HBP)=P(HBP)\times P(\text{Exercise}\geq90|HBP)=0.28\times0.32 = 0.0896\approx9\%\) (Wait, no, maybe I mixed up. Wait, the problem says "86% of those who do not have high blood pressure exercise at least 90 minutes per week", so the relative frequency for "Does not have HBP" and "Exercise \(\geq90\)" is \(86\%\) of the "Does not have HBP" total. The "Does not have HBP" total is \(72\%\) of the population. So the relative frequency (as a percentage of total population) for "Does not have HBP" and "Exercise \(\geq90\)" is \(0.72\times0.86 = 0.6192\approx62\%\), and for "Has HBP" and "Exercise \(\geq90\)" is \(0.28\times0.32 = 0.0896\approx9\%\)? Wait, no, the "of those with HBP" who exercise \(\geq90\) is \(32\%\), so the relative frequency (percentage of total) for "Has HBP" and "Exercise \(\geq90\)" is \(28\%\times32\% = 8.96\%\approx9\%\), and for "Has HBP" and "Exercise < 90" is \(28\%\times(1 - 0.32)=28\%\times0.68 = 19.04\%\approx19\%\)? Wait, no, maybe the table is a two - way relative frequency table. Let's check the totals.

The total for "Has HBP" is \(28\%\), "Does not have HBP" is \(72\%\).

For "Exercise \(\geq90\)":
- "Does not have HBP": \(72\%\times86\% = 61.92\%\approx62\%\)
- "Has HBP": \(28\%\times32\% = 8.96\%\approx9\%\)
- So total for "Exercise \(\geq90\)": \(62 + 9=71\%\)

For "Exercise < 90":
- "Does not have HBP": \(72\%\times(1 - 86\%)=72\%\times14\% = 10.08\%\approx10\%\)
- "Has HBP": \(28\%\times(1 - 32\%)=28\%\times68\% = 19.04\%\approx19\%\)
- Total for "Exercise < 90": \(10+19 = 29\%\)? Wait, no, \(10 + 19=29\)? Wait, \(10+19 = 29\), but \(71+29 = 100\). Wait, but let's check the tables.

Looking at the third table (from the left? Wait, the tables are:

First table: Exercise \(\geq90\) for HBP:24%, \( eg\)HBP:23%... No.

Wait, let's recast. The correct relative frequencies should be:

- For "Has HBP" row:
- Exercise \(\geq90\): \(P(HBP\cap\text{Exercise}\geq90)=P(HBP)\times P(\text{Exercise}\geq90|HBP)=0.28\times0.32 = 0.0896\approx9\%\)
- Exercise < 90: \(P(HBP\cap\text{Exercise}<90)=P(HBP)\times P(\text{Exercise}<90|HBP)=0.28\times(1 - 0.32)=0.28\times0.68 = 0.1904\approx19\%\)
- For "Does not have HBP" row:
- Exercise \(\geq90\): \(P( eg HBP\cap\text{Exercise}\geq90)=P( eg HBP)\times P(\text{Exercise}\geq90| eg HBP)=0.72\times0.86 = 0.6192\approx62\%\)
- Exercise < 90: \(P( eg HBP\cap\text{Exercise}<90)=P( eg HBP)\times P(\text{Exercise}<90| eg HBP)=0.72\times(1 - 0.86)=0.72\times0.14 = 0.1008\approx10\%\)
- Totals:
- Exercise \(\geq90\): \(9 + 62=71\%\)
- Exercise < 90: \(19+10 = 29\%\)
- Has HBP: \(19 + 9=28\%\)
- Does not have HBP: \(10 + 62=72\%\)

Looking at the tables, the third table (counting from left, the one with "Has high blood pressure" row: 9% (Exercise \(\geq90\)), 19% (Exercise < 90); "Does not have high blood pressure" row:62% (Exercise \(\geq90\)), 10% (Exercise < 90); totals 71%, 29%, 28%, 72%) matches these calculations.

Step2: Match with tables

We calculate the expected relative frequencies and match with the given tables. The table with:
- Has high blood pressure: 9% (Exercise \(\geq90\)), 19% (Exercise < 90)
- Does not have high blood pressure:62% (Exercise \(\geq90\)), 10% (Exercise < 90)
- Totals:71% (Exercise \(\geq90\)), 29% (Exercise < 90); 28% (Has HBP), 72% (Does not have HBP) is the correct one. Looking at the image, this is the third table (middle - right? Wait, the third table from the left in the image as per the user's provided image structure).

Answer:

The third relative frequency table (the one with "Has high blood pressure" row: 9%, 19%; "Does not have high blood pressure" row: 62%, 10%; totals 71%, 29%, 28%, 72%)