Step-by-Step Guidance for Statistics Midterm Practice Exam

SECTION 1: DATA ANALYSIS (Chapters 1 & 2)

Problem 1

A researcher records the number of hours 8 students slept the night before an exam: 3, 5, 6, 7, 7, 8, 9, 12

Q1. Calculate the Mean (\(\bar{x}\)) and Median (M).

Background

Topic: Measures of Central Tendency

This question tests your ability to compute the mean and median, which are two common ways to describe the center of a data set.

Key Terms and Formulas:

• Mean (\(\bar{x}\)): The arithmetic average of the data.

• Median (M): The middle value when the data are ordered.

Mean formula:

$\bar{x} = \frac{\sum x_i}{n}$

Median: Arrange the data in order and find the middle value (or average the two middle values if n is even).

Step-by-Step Guidance

1. Order the data from smallest to largest: 3, 5, 6, 7, 7, 8, 9, 12.

2. Count the number of data points (n = 8).

3. To find the mean, add all the values together and divide by 8.

4. To find the median, since n is even, average the 4th and 5th values in the ordered list.

Try solving on your own before revealing the answer!

Final Answer:

Mean: 7.125, Median: 7

The mean is the sum of all values divided by 8. The median is the average of the 4th and 5th values (both 7).

Q2. Calculate the 5-Number Summary: (Min, Q_1, Med, Q_3, Max).

Background

Topic: Descriptive Statistics – Five-Number Summary

This question tests your ability to summarize a data set using the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

Key Terms and Formulas:

• Minimum: Smallest value

• Q1: Median of the lower half (not including the median if n is even)

• Median: Middle value

• Q3: Median of the upper half

• Maximum: Largest value

Step-by-Step Guidance

1. List the data in order: 3, 5, 6, 7, 7, 8, 9, 12.

2. Identify the minimum and maximum values.

3. Find the median (already done above).

4. Find Q1: Median of the lower half (first 4 values).

5. Find Q3: Median of the upper half (last 4 values).

Try solving on your own before revealing the answer!

Final Answer:

Five-number summary: 3, 5.5, 7, 8.5, 12

Q1 is the average of 5 and 6; Q3 is the average of 8 and 9.

Q3. Find the IQR (Interquartile Range).

Background

Topic: Measures of Spread

This question tests your ability to calculate the interquartile range, which measures the spread of the middle 50% of the data.

Key Terms and Formulas:

• IQR = Q3 - Q1

Step-by-Step Guidance

1. Recall your values for Q1 and Q3 from the previous question.

2. Subtract Q1 from Q3 to find the IQR.

Try solving on your own before revealing the answer!

Final Answer:

IQR = 3

Subtract Q1 (5.5) from Q3 (8.5).

Q4. Identify Outliers: Use the 1.5 × IQR rule. Is the value 12 a suspected outlier? Show your work.

Background

Topic: Outlier Detection

This question tests your ability to use the 1.5 × IQR rule to determine if a data point is an outlier.

Key Terms and Formulas:

• Lower bound: Q1 - 1.5 × IQR

• Upper bound: Q3 + 1.5 × IQR

• If a value is outside these bounds, it is a suspected outlier.

Step-by-Step Guidance

1. Calculate 1.5 × IQR.

2. Add this value to Q3 to get the upper bound.

3. Subtract this value from Q1 to get the lower bound.

4. Check if 12 is greater than the upper bound or less than the lower bound.

Try solving on your own before revealing the answer!

Final Answer:

Upper bound: 13; 12 is not an outlier since it is less than 13.

Using the 1.5 × IQR rule, 12 is within the acceptable range.

Q5. Interpret the Standard Deviation: If the standard deviation is s = 2.67, write one sentence explaining what this means in the context of student sleep.

Background

Topic: Interpreting Standard Deviation

This question tests your ability to interpret the meaning of standard deviation in context.

Key Terms:

• Standard deviation (s): A measure of how spread out the data values are from the mean.

Step-by-Step Guidance

1. Recall that standard deviation tells you, on average, how far each data point is from the mean.

2. Think about what this means for the number of hours students slept.

3. Write a sentence that connects the value of s = 2.67 to the context of student sleep.

Try writing your own interpretation before revealing the answer!

Final Answer:

On average, the number of hours students slept differs from the mean by about 2.67 hours.

SECTION 2: THE NORMAL DISTRIBUTION (Chapter 3)

Problem 2

The scores on a national exam are normally distributed with a mean (\(\mu\)) of 75 and a standard deviation (\(\sigma\)) of 10. N(75, 10).

Q1. The 68-95-99.7 Rule: Approximately what percentage of students scored between 55 and 95?

Background

Topic: Empirical Rule (68-95-99.7 Rule)

This question tests your understanding of the normal distribution and the empirical rule for standard deviations from the mean.

Key Terms and Formulas:

• Empirical Rule: In a normal distribution, about 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3.

• \(\mu\): Mean

• \(\sigma\): Standard deviation

Step-by-Step Guidance

1. Calculate how many standard deviations 55 and 95 are from the mean (75).

2. Recognize that 55 is 2 standard deviations below the mean, and 95 is 2 above.

3. Apply the empirical rule to estimate the percentage of students in this range.

Try solving on your own before revealing the answer!

Final Answer:

Approximately 95% of students scored between 55 and 95.

Q2. Standardizing: A student scored an 88. Calculate their Z-score.

Background

Topic: Z-scores and Standardization

This question tests your ability to convert a raw score to a Z-score, which tells you how many standard deviations a value is from the mean.

Key Terms and Formulas:

• Z-score formula: $z = \frac{x - \mu}{\sigma}$

Step-by-Step Guidance

1. Identify the values: $x = 88$, $\mu = 75$, $\sigma = 10$.

2. Plug these values into the Z-score formula.

3. Simplify the numerator and denominator before dividing.

Try solving on your own before revealing the answer!

Final Answer:

Z = 1.3

The student's score is 1.3 standard deviations above the mean.

Q3. Using Table A: What proportion of students scored higher than an 88? (Show the subtraction from 1).

Background

Topic: Normal Distribution and Table Lookup

This question tests your ability to use the standard normal table (Table A) to find the area to the right of a given Z-score.

Key Terms and Formulas:

• Table A: Standard normal cumulative probability table

• Area to the right: $1 - P(Z < z)$

Step-by-Step Guidance

1. Use your Z-score from the previous question (1.3).

2. Look up the area to the left of Z = 1.3 in Table A.

3. Subtract this value from 1 to get the area to the right.

Try solving on your own before revealing the answer!

Final Answer:

Proportion higher than 88: 0.0968

1 - 0.9032 = 0.0968

Q4. Inverse Calculation: What score would a student need to be in the top 5% (95th percentile) of test-takers? (Use your calculator invNorm or Table A).

Background

Topic: Inverse Normal Calculations

This question tests your ability to find a raw score corresponding to a given percentile in a normal distribution.

Key Terms and Formulas:

• Percentile: The value below which a given percentage of observations fall.

• Inverse normal calculation: $x = \mu + z^* \sigma$

• Use Table A or invNorm to find $z^*$ for the 95th percentile.

Step-by-Step Guidance

1. Find the Z-score that corresponds to the 95th percentile (top 5%).

2. Plug this Z-score, along with the mean and standard deviation, into the formula $x = \mu + z^* \sigma$.

3. Multiply $z^*$ by $\sigma$ and add to $\mu$.

Try solving on your own before revealing the answer!

Final Answer:

Score needed: 91.45

Used $z^* \approx 1.645$; $x = 75 + (1.645 \times 10)$

SECTION 3: PROBABILITY & SAMPLING (Chapter 8)

Problem 3

A large bag of marbles contains 30% Red, 20% Blue, and 50% Green marbles.

Q1. If you pick one marble at random, what is the probability it is not Green?

Background

Topic: Basic Probability

This question tests your understanding of probability and complementary events.

Key Terms and Formulas:

• Probability of an event: $P(A)$

• Complement rule: $P(\text{not Green}) = 1 - P(\text{Green})$

Step-by-Step Guidance

1. Identify the probability of picking a Green marble (given as 0.50).

2. Use the complement rule to find the probability of not Green.

Try solving on your own before revealing the answer!

Final Answer:

Probability not Green: 0.50

1 - 0.50 = 0.50

Q2. If you pick two marbles (with replacement), what is the probability they are both Red?

Background

Topic: Probability of Independent Events

This question tests your ability to calculate the probability of two independent events both occurring.

Key Terms and Formulas:

• With replacement: Each pick is independent.

• Probability of both: $P(A \text{ and } B) = P(A) \times P(B)$

Step-by-Step Guidance

1. Find the probability of picking a Red marble on the first draw (0.30).

2. Since the draws are independent, multiply the probability by itself for the second draw.

Try solving on your own before revealing the answer!

Final Answer:

Probability both Red: 0.09

0.30 × 0.30 = 0.09

Q3. Parameter vs. Statistic: You take a sample of 50 marbles and find that 28% are Red. Identify which number is the parameter (p) and which is the statistic (\(\hat{p}\)).

Background

Topic: Parameters vs. Statistics

This question tests your understanding of the difference between a population parameter and a sample statistic.

Key Terms:

• Parameter (p): A value that describes a characteristic of the entire population.

• Statistic (\(\hat{p}\)): A value that describes a characteristic of a sample.

Step-by-Step Guidance

1. Identify the percentage of Red marbles in the population (given as 30%).

2. Identify the percentage found in the sample (28%).

3. Label each value as either the parameter or the statistic.

Try labeling them before revealing the answer!

Final Answer:

Parameter p = 0.30; Statistic \(\hat{p}\) = 0.28

SECTION 4: EXPERIMENTAL DESIGN (Chapter 9)

Problem 4

A medical study wants to test the effectiveness of a new blood pressure medication. There are 100 volunteers: 60 over the age of 50, and 40 under the age of 50.

Q1. Identify the variables: What is the Explanatory Variable and what is the Response Variable?

Background

Topic: Variables in Experimental Design

This question tests your ability to distinguish between explanatory (independent) and response (dependent) variables in an experiment.

Key Terms:

• Explanatory Variable: The variable manipulated by the researcher (the cause).

• Response Variable: The outcome measured (the effect).

Step-by-Step Guidance

1. Identify what is being changed or controlled by the researchers (the treatment).

2. Identify what is being measured as the outcome.

Try identifying the variables before revealing the answer!

Final Answer:

Explanatory: Medication; Response: Blood Pressure

Q2. Randomization: Use the random digits below to select the first two subjects from a group labeled 01–40.

Line 105: 95592 94007 69971 91481 60779

Background

Topic: Random Sampling

This question tests your ability to use random digits to select subjects from a numbered list.

Key Terms and Process:

• Random digits: Use pairs of digits to represent subject numbers (01–40).

• Skip numbers outside the range or repeats.

Step-by-Step Guidance

1. Break the random digit string into pairs: 95, 59, 29, 40, etc.

2. Ignore any pairs not in the range 01–40.

3. Select the first two valid numbers as your subjects.

Try selecting the subjects before revealing the answer!

Final Answer:

Selected: 29 and 40

First two valid pairs in the range 01–40.

Q3. Design Comparison: Explain how you would implement a Block Design using "Age" as the blocking variable. Why is a Block Design better here than a Completely Randomized Design?

Background

Topic: Experimental Design – Blocking

This question tests your understanding of block design and its advantages in experiments.

Key Terms:

• Block Design: Grouping subjects by a variable (like age) before random assignment.

• Completely Randomized Design: Randomly assign all subjects without blocking.

Step-by-Step Guidance

1. Describe how to split subjects into blocks based on age.

2. Explain how to randomly assign treatments within each block.

3. Discuss why blocking by age could improve the experiment's validity.

Try explaining the design before revealing the answer!

Final Answer:

Block by age (under 50, over 50), then randomly assign treatments within each block. This reduces variation due to age.

Q4. Blinding: Explain the difference between a Single-Blind and Double-Blind experiment in this context.

Background

Topic: Blinding in Experiments

This question tests your understanding of how blinding reduces bias in experiments.

Key Terms:

• Single-Blind: Subjects do not know which treatment they receive.

• Double-Blind: Neither subjects nor experimenters know who receives which treatment.

Step-by-Step Guidance

1. Define single-blind in the context of this medication study.

2. Define double-blind in the same context.

3. Explain why double-blind is generally preferred.

Try explaining the difference before revealing the answer!

Final Answer:

Single-blind: Subjects don't know their group; double-blind: neither subjects nor those measuring blood pressure know.

SECTION 5: CALCULATOR SKILLS (TI-84 CE)

Q1. Which menu do you use to find the mean and standard deviation after entering data in a list?

Background

Topic: Calculator Data Analysis

This question tests your knowledge of TI-84 calculator functions for descriptive statistics.

Key Terms:

• 1-Var Stats: The menu for one-variable statistics.

Step-by-Step Guidance

1. Enter your data into a list (e.g., L1).

2. Access the STAT menu, then CALC, and select 1-Var Stats.

Try recalling the menu before revealing the answer!

Final Answer:

1-Var Stats

Q2. Which command finds the area under the curve between two Z-scores?

Background

Topic: Calculator Probability Functions

This question tests your knowledge of TI-84 commands for normal distributions.

Key Terms:

• normalcdf: The command for cumulative area between two values.

Step-by-Step Guidance

1. Identify the lower and upper Z-scores for your interval.

2. Use the normalcdf command with these values.

Try recalling the command before revealing the answer!

Final Answer:

normalcdf

Q3. If you are given an area of 0.25 (the 25th percentile) and asked to find the Z-score, which command do you use?

Background

Topic: Calculator Inverse Normal

This question tests your knowledge of finding Z-scores from percentiles using the TI-84.

Key Terms:

• invNorm: The command for finding Z-scores from areas/percentiles.

Step-by-Step Guidance

1. Recognize that you need the Z-score for a given area to the left (0.25).

2. Use the invNorm command with this area.

Try recalling the command before revealing the answer!

Final Answer:

invNorm