Practice Problems

Problems:

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Problem 1

Explanation:

When you want to know how much time was spent exercising on average, the mean helps you figure that out. Let's add up all the times and then spread them evenly across the days to find the average.

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Problem 2

Explanation: The median is like finding the middle of a list. If you arrange all the scores from the lowest to the highest, the median is the one sitting right in the center.

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Problem 3

Explanation:

The mode is the number that pops up the most often. It's like finding the most popular item sold during the week. Let's see which number shows up the most!

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Problem 4

Explanation:

The range shows how much the delays vary from day to day. We look at the biggest and smallest delays to see the difference between them.

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Problem 5

Explanation: Sometimes, we need to look at data in different ways: the mean (average), the median (middle value), and the mode (most common value). Each one tells us something different about how much time was spent studying.

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Problem 6

Explanation:

The IQR helps us understand how spread out the middle half of the data is. It's like finding the range, but for the middle part of our list.

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Solutions:

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Problem 1

Solution:

• Step 1: Add the times together.
  • Why? The mean is the total amount of time spread equally over the days, so we need to find the total first.
  • $30 + 45 + 35 + 50 + 40 + 60 + 55 = :highlight[315]$
• Step 2: Count how many days there are.
  • Why? We divide the total by the number of days to find the mean.
  • There are 7 days.
• Step 3: Divide the total by the number of days.
  • Why? Dividing the total time by the number of days gives the mean, or average time exercised each day.
  • $315 \div 7 = :success[45]$ Answer**:** The mean time exercised is 45 minutes.

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Problem 2

Solution:

• Step 1: Arrange the scores in order from smallest to largest.
  • Why? The median is the middle score, so we need to put the scores in order first.
  • Ordered scores: $68, 72, 78, 85, 88, 90, 95$
• Step 2: Identify the middle score.
  • Why? The middle score is the median, showing the central value in the dataset.
  • The middle score is 85. Answer**:** The median score is 85.

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Problem 3

Solution:

• Step 1: Count how often each number appears.
  • Why? The mode is the number that appears most often, so we need to find the frequency of each number.
  • 20 appears 2 times, 22 appears 3 times, 25 appears 1 time, 19 appears 1 time.
• Step 2: Identify the number that appears most often.
  • Why? The number that appears the most is the mode, showing the most common sales number.
  • The number 22 appears most frequently. Answer**:** The mode is 22 items.

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Problem 4

Solution:

• Step 1: Identify the highest delay.
  • Why? The range measures how much the delays vary, so we start by finding the highest delay.
  • The highest delay is 15 minutes.
• Step 2: Identify the lowest delay.
  • Why? We also need to know the lowest delay to calculate the difference between the highest and lowest values.
  • The lowest delay is 0 minutes.
• Step 3: Subtract the lowest delay from the highest delay.
  • Why? Subtracting gives us the difference between the highest and lowest values, which is the range.
  • $15 - 0 = :success[15]$ Answer**:** The range of train delays is 15 minutes.

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Problem 5

Solution:

Step 1: Find the Mean

• Step 1: Add the study hours together.
  • Why? To find the mean, we first need the total sum of all the hours.
  • $4 + 6 + 5 + 7 + 8 + 6 + 4 + 9 = :highlight[49]$
• Step 2: Count how many students there are.
  • Why? Knowing the number of students allows us to divide the total hours evenly.
  • There are 8 students.
• Step 3: Divide the total by the number of students.
  • Why? Dividing gives us the mean, which is the average time spent studying.
  • $49 \div 8 = :highlight[6.125]$ Answer**:** The mean study time is 6.13 hours (rounded to two decimal places).

Step 2: Find the Median

• Step 1: Arrange the study hours in order from smallest to largest.
  • Why? The median is the middle value, so we need to arrange the hours in order first.
  • Ordered hours: $4, 4, 5, 6, 6, 7, 8, 9$
• Step 2: Identify the two middle numbers.
  • Why? When there's an even number of values, the median is the average of the two middle numbers.
  • The two middle numbers are 6 and 6.
• Step 3: Find the average of the two middle numbers.
  • Why? The average of these two numbers gives us the median.
  • $\frac{6 + 6}{2} = :success[6]$ Answer**:** The median study time is 6 hours.

Step 3: Find the Mode

• Step 1: Identify the number that appears most often.
  • Why? The mode is the number that occurs most frequently, showing the most common study time.
  • The numbers 4 and 6 both appear 2 times. Answer**:** The modes are 4 hours and 6 hours (bimodal distribution).

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Problem 6

Solution:

Step 1: Arrange the numbers in order (already done).

• Why? The IQR is based on the quartiles, which divide the ordered data into four equal parts. Step 2: Split the data into two halves.

• Why? We need to find the quartiles, so we divide the data into lower and upper halves.

• Lower half: $1, 3, 5, 7$

• Upper half: $9, 10, 12, 15$ Step 3: Find Q1 (the first quartile).

• Why? Q1 is the median of the lower half, showing the value that separates the lowest 25% of the data from the rest.

• The median of $1, 3, 5, 7$ is between $3$ and $5$, so $Q1 = \frac{3 + 5}{2} = :highlight[4]$. Step 4: Find Q3 (the third quartile).

• Why? Q3 is the median of the upper half, showing the value that separates the highest 25% of the data from the rest.

• The median of $9, 10, 12, 15$ is between $10$ and $12$, so $Q3 = \frac{10 + 12}{2} = :highlight[11]$. Step 5: Calculate the IQR.

• Why? Subtracting Q1 from Q3 gives us the IQR, which measures the spread of the middle 50% of the data.

• $11 - 4 = :success[7]$ Answer**:** The IQR is 7.

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