PSAT/NMSQT Math: Problem and Data Analysis Questions and Answers 1 — Questions and Answers
Question 1: The two-way table below shows the results of a survey of 200 high school students regarding their primary mode of transportation to school, categorized by grade level. | | Bus | Car | Walk | Total | |---|---|---|---|---| | Junior | 45 | 30 | 15 | 90 | | Senior | 25 | 70 | 15 | 110 | | Total | 70 | 100 | 30 | 200 | Of the students who ride the bus to school, what percentage are juniors? (Round to the nearest tenth of a percent)
- 22.5%
- 45.0%
- 64.3% (Correct answer)
- 70.0%
Correct answer: 64.3%
The question asks for the percentage of a specific subgroup: juniors among bus riders. First, identify the total number of students who ride the bus, which is 70. Then, find the number of juniors within that group, which is 45. To find the percentage, divide the part (juniors who ride the bus) by the whole (all students who ride the bus) and multiply by 100: (45 / 70) * 100 ≈ 64.3%.
Question 2: A biologist is studying the growth of a certain plant species. The scatterplot for the plant's height over time shows a positive linear association. The line of best fit is given by the equation y = 1.8x + 5.2, where y is the height of the plant in centimeters and x is the number of weeks since the study began. What is the best interpretation of the slope of the line of best fit?
- The plant's predicted initial height at the start of the study was 1.8 cm.
- For each additional week, the plant's predicted height increases by 1.8 cm. (Correct answer)
- For each additional week, the plant's predicted height increases by 5.2 cm.
- The plant's predicted initial height at the start of the study was 5.2 cm.
Correct answer: For each additional week, the plant's predicted height increases by 1.8 cm.
In a linear equation of the form y = mx + b, 'm' represents the slope. The slope describes the rate of change, meaning how much the y-variable changes for a one-unit increase in the x-variable. In this context, y is the height and x is the number of weeks. Therefore, the slope of 1.8 means that for every additional week, the plant's height is predicted to increase by 1.8 cm.
Question 3: The list below shows the number of goals scored by a soccer player in 9 consecutive games: 2, 3, 0, 1, 3, 4, 2, 1, 3 If the player scores 10 goals in the tenth game, which of the following statistical measures of the data will change the most?
- Mean
- Median
- Mode
- Range (Correct answer)
Correct answer: Range
The new data point, 10, is an outlier. Let's analyze the effect on each measure. The original data set is {0, 1, 1, 2, 2, 3, 3, 3, 4}. The original range is 4 - 0 = 4. The new data set is {0, 1, 1, 2, 2, 3, 3, 3, 4, 10}. The new range is 10 - 0 = 10. The change in range is 6. The original mean is 19/9 ≈ 2.11 and the new mean is 29/10 = 2.9, a change of about 0.79. The original median is 2 and the new median is (2+3)/2 = 2.5, a change of 0.5. The mode (3) does not change. The range is affected most significantly by extreme values.
Question 4: A map of a state park is drawn to scale. The legend indicates that 2 inches on the map represents 5 miles in the park. Two trailheads are 7 inches apart on the map. What is the actual distance between the two trailheads in miles?
- 17.5 miles (Correct answer)
- 15.5 miles
- 14.0 miles
- 20.0 miles
Correct answer: 17.5 miles
This problem can be solved using a proportion. Let x be the actual distance in miles. The relationship is (2 inches / 5 miles) = (7 inches / x miles). To solve for x, you can cross-multiply: 2x = 5 * 7, which gives 2x = 35. Dividing both sides by 2 gives x = 17.5 miles.
Question 5: A technology blogger wants to determine the most popular brand of smartphone among young adults in a particular city. The blogger conducts a survey by asking the first 100 people who visit their tech blog on a Monday morning to name their smartphone brand. Why might the conclusion drawn from this survey NOT be reliable for the city's entire young adult population?
- The sample size of 100 people is too small to draw any conclusions.
- The survey was conducted on a Monday, when people are typically busy.
- The sample is likely biased because it is not representative of all young adults in the city. (Correct answer)
- The survey should have been conducted in person instead of online.
Correct answer: The sample is likely biased because it is not representative of all young adults in the city.
The method used is a convenience sample, which surveys a group that is easy to reach. Visitors to a specific technology blog are likely more interested in technology than the general population and may have different brand preferences. This creates a sampling bias, meaning the sample is not representative of all young adults in the city. Therefore, the conclusions cannot be reliably generalized to that larger population.
Question 6: A company that manufactures light bulbs tests a random sample of 500 bulbs from a production batch of 20,000 bulbs. The test finds that 8 of the bulbs are defective. Based on this sample, what is the best estimate for the total number of defective bulbs in the entire batch?
- 128
- 320 (Correct answer)
- 400
- 1,600
Correct answer: 320
First, calculate the rate of defective bulbs in the sample: 8 defective bulbs / 500 total bulbs = 0.016. This means 1.6% of the bulbs in the sample are defective. To estimate the total number of defective bulbs in the batch, multiply this rate by the total number of bulbs in the batch: 0.016 * 20,000 = 320. Alternatively, set up a proportion: (8/500) = (x/20000). Solving for x gives x = 320.
PSAT/NMSQT Math: Problem and Data Analysis Questions and Answers 1
The two-way table below shows the results of a survey of 200 high school students regarding their primary mode of transportation to school, categorized by grade level.
| | Bus | Car | Walk | Total |
|---|---|---|---|---|
| Junior | 45 | 30 | 15 | 90 |
| Senior | 25 | 70 | 15 | 110 |
| Total | 70 | 100 | 30 | 200 |
Of the students who ride the bus to school, what percentage are juniors? (Round to the nearest tenth of a percent)