Step-by-Step Guidance for MTH 243 Exam 1 Review (Statistics)

Q1. Determine whether the given values are from a discrete or continuous data set.

• a. In the Literary Digest poll, Landon received 16,679,583 votes.

• b. The volume of cola in a can of regular Coke is 12.3 oz.

Background

Topic: Types of Data (Discrete vs. Continuous)

This question tests your understanding of the difference between discrete and continuous data in statistics.

Key Terms:

• Discrete data: Data that can only take specific, separate values (often counts).

• Continuous data: Data that can take any value within a range (often measurements).

Step-by-Step Guidance

1. For each value, ask: Can this value be broken down into smaller parts, or does it only make sense as whole numbers?

2. Consider if the value is a count (like number of people or objects) or a measurement (like weight, volume, or time).

3. Decide if the value could, in theory, take on any value within a range (continuous) or only specific values (discrete).

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Q2. Determine which of the four levels of measurement is most appropriate.

• a. Types of movies (drama, comedy, adventure, documentary, etc.)

• b. Ranks of cars evaluated by Consumer’s Union.

• c. Measured amounts of greenhouse gases emitted by cars.

Background

Topic: Levels of Measurement

This question tests your ability to classify data as nominal, ordinal, interval, or ratio.

Key Terms:

• Nominal: Categories with no order (e.g., types of movies).

• Ordinal: Categories with a meaningful order, but differences between values are not meaningful (e.g., ranks).

• Interval: Ordered, equal intervals, but no true zero (e.g., temperature in Celsius).

• Ratio: Ordered, equal intervals, and a true zero (e.g., height, weight, measured amounts).

Step-by-Step Guidance

1. For each example, ask: Is there a natural order? Are differences meaningful? Is there a true zero?

2. Match each example to the correct level of measurement based on these criteria.

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Q3. Identify the following as either observational or experimental:

• a. A Gallup poll surveyed 1018 adults by telephone, and 22% of them reported that they had smoked cigarettes within the past week.

• b. Children with 337 upper respiratory tract infections were given Echinacea, and children with 370 of the infections were given placebos.

Background

Topic: Types of Studies (Observational vs. Experimental)

This question tests your ability to distinguish between observational studies and experiments.

Key Terms:

• Observational study: Researchers observe outcomes without assigning treatments.

• Experimental study: Researchers assign treatments and observe effects.

Step-by-Step Guidance

1. For each scenario, determine if the researchers assigned a treatment or simply observed existing conditions.

2. If a treatment or intervention was assigned, it is experimental; if not, it is observational.

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Q4. Identify which of these types of sampling is used: random, stratified, systematic, cluster, convenience.

• a. A student collected measurements of arm lengths from her family members.

• b. On the day of the last presidential election, ABC News organized an exit poll in which specific polling stations were randomly selected and all voters were surveyed as they left the premises.

• c. Every fifth driver was stopped and interviewed at a sobriety checkpoint.

Background

Topic: Sampling Methods

This question tests your understanding of different sampling techniques used in statistics.

Key Terms:

• Random sampling: Every member has an equal chance of being selected.

• Stratified sampling: Population divided into subgroups, then random samples taken from each subgroup.

• Systematic sampling: Every nth member is selected.

• Cluster sampling: Population divided into groups (clusters), some clusters are randomly selected, and all members of those clusters are surveyed.

• Convenience sampling: Samples are taken from a group that is easy to access.

Step-by-Step Guidance

1. For each scenario, identify how the sample was chosen: Was it random, based on a pattern, from a subgroup, or just convenient?

2. Match the description to the correct sampling method using the definitions above.

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Q5. Identify the class width, class midpoints, and class boundaries of the following frequency distribution:

Tar (mg) in filter cigarettes	Frequency
2-5	2
6-9	2
10-13	6
14-17	15

Background

Topic: Frequency Distributions

This question tests your ability to interpret and analyze frequency tables, specifically class width, midpoints, and boundaries.

Key Terms and Formulas:

• Class width: $\text{Class width} = \text{Upper limit of a class} - \text{Lower limit of the same class} + 1$ (or just the difference between lower limits of consecutive classes).

• Class midpoint: $\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

• Class boundaries: The values that separate classes, usually halfway between the upper limit of one class and the lower limit of the next.

Step-by-Step Guidance

1. Find the class width by subtracting the lower limit of the first class from the lower limit of the second class.

2. Calculate the midpoint for each class using the formula above.

3. Determine the class boundaries by finding the average between the upper limit of one class and the lower limit of the next class.

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Q6. Construct a relative and a cumulative frequency distribution of the Frequency distribution in question 5 above.

Background

Topic: Frequency Distributions (Relative and Cumulative)

This question tests your ability to convert a frequency distribution into relative and cumulative forms.

Key Terms and Formulas:

• Relative frequency: $\text{Relative frequency} = \frac{\text{Frequency of a class}}{\text{Total frequency}}$

• Cumulative frequency: The sum of the frequencies for that class and all previous classes.

Step-by-Step Guidance

1. Calculate the total frequency by adding up all the frequencies from question 5.

2. For each class, divide its frequency by the total frequency to get the relative frequency.

3. For cumulative frequency, start with the first class and add each subsequent class's frequency to the running total.

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Q7. A data set consists of heights of 100 randomly selected adults and the heights range from 53.5 in. to 83.4 in. You wish to construct a frequency table with 10 classes. What is the most suitable upper and lower limits for the first class of the frequency table?

Background

Topic: Constructing Frequency Tables

This question tests your ability to determine class limits when constructing a frequency table.

Key Terms and Formulas:

• Range: $\text{Range} = \text{Maximum value} - \text{Minimum value}$

• Class width: $\text{Class width} = \frac{\text{Range}}{\text{Number of classes}}$ (usually round up to a convenient number)

Step-by-Step Guidance

1. Calculate the range of the data set using the maximum and minimum values.

2. Divide the range by the number of classes (10) to find the class width, rounding up if necessary.

3. Set the lower limit of the first class at or just below the minimum value, and add the class width to find the upper limit.

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Q8. The FICO credit rating scores obtained in a simple random sample are listed below. Find the mean, median, mode, midrange, standard deviation, and variance.

Scores: 714, 751, 664, 789, 818, 779, 698, 836, 753, 834, 693, 802

Background

Topic: Descriptive Statistics

This question tests your ability to compute various measures of central tendency and variability.

Key Terms and Formulas:

• Mean: $\bar{x} = \frac{\sum x_i}{n}$

• Median: The middle value when data are ordered.

• Mode: The value(s) that appear most frequently.

• Midrange: $\text{Midrange} = \frac{\text{Maximum} + \text{Minimum}}{2}$

• Standard deviation (sample): $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

• Variance (sample): $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$

Step-by-Step Guidance

1. Order the data from smallest to largest to help find the median and midrange.

2. Calculate the mean by summing all values and dividing by the number of values.

3. Find the median by locating the middle value(s) in the ordered list.

4. Identify the mode by finding the value(s) that appear most often.

5. Compute the midrange by averaging the maximum and minimum values.

6. Calculate the standard deviation and variance using the formulas above, starting with the mean and then finding squared deviations.

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Q9. Find the mean amount and standard deviation, s, of tar in non-filtered cigarettes, and find the coefficient of variation of s.

Tar (mg) in Non-filtered Cigarettes	Frequency
10-13	1
14-17	0
18-21	15
22-25	7
26-29	2

Background

Topic: Grouped Data Statistics

This question tests your ability to compute mean, standard deviation, and coefficient of variation from grouped data.

Key Terms and Formulas:

• Mean (for grouped data): $\bar{x} = \frac{\sum (f \cdot x_m)}{\sum f}$, where $x_m$ is the class midpoint.

• Standard deviation (for grouped data): $s = \sqrt{\frac{\sum f(x_m - \bar{x})^2}{\sum f - 1}}$

• Coefficient of variation: $CV = \frac{s}{\bar{x}} \times 100\%$

Step-by-Step Guidance

1. Find the midpoint for each class interval.

2. Multiply each midpoint by its frequency and sum these products to find the numerator for the mean.

3. Divide by the total frequency to get the mean.

4. For standard deviation, calculate each squared deviation from the mean, multiply by frequency, sum, and divide by (total frequency - 1), then take the square root.

5. Calculate the coefficient of variation using the mean and standard deviation.

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Q10. The mean of a set of data is 0.88 and its standard deviation is 1.73. Find the z score for a value of 3.96. Round results to the nearest hundredth.

Background

Topic: Z-Scores (Standard Scores)

This question tests your ability to calculate a z-score, which measures how many standard deviations a value is from the mean.

Key Formula:

• $z = \frac{x - \mu}{\sigma}$ (for population) or $z = \frac{x - \bar{x}}{s}$ (for sample)

Step-by-Step Guidance

1. Identify the value ($x$), mean ($\bar{x}$), and standard deviation ($s$) from the problem.

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

3. Subtract the mean from the value, then divide by the standard deviation.

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Q11. Determine which score corresponds to the higher relative position: a score of 65 on a test with a mean of 87 and a standard deviation of 21, or a score of 515 on a test with a mean of 737 and a standard deviation of 198?

Background

Topic: Comparing Relative Positions Using Z-Scores

This question tests your ability to compare scores from different distributions using z-scores.

Key Formula:

• $z = \frac{x - \mu}{\sigma}$

Step-by-Step Guidance

1. Calculate the z-score for each test score using the provided means and standard deviations.

2. Compare the two z-scores to determine which is higher (i.e., which score is further above its mean).

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Q12. The following is the number of points scored in the Super Bowl for a recent period of 24 years.

36, 37, 37, 39, 39, 41, 43, 44, 44, 47, 50, 53, 54, 55, 56, 56, 57, 59, 61, 61, 65, 69, 69, 75

• a. Find the location of $P_{32}$.

• b. Find the location of $P_{80}$.

Background

Topic: Percentiles

This question tests your ability to find the position (location) of a given percentile in a data set.

Key Formula:

• $L = \frac{k}{100} \times n$, where $L$ is the location, $k$ is the percentile, and $n$ is the number of data values.

Step-by-Step Guidance

1. Count the total number of data values ($n$).

2. Plug the percentile ($k$) and $n$ into the formula to find the location for each percentile.

3. Interpret the location: If $L$ is not a whole number, round up to the next integer to find the position in the ordered data set.

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Q13. Construct a boxplot for the data given in the stem-and-leaf plot. Include values of the 5-number summary in your boxplot. (Use the number-line for your horizontal scale.)

Stem-and-leaf plot provided.

Background

Topic: Boxplots and Five-Number Summary

This question tests your ability to extract data from a stem-and-leaf plot, compute the five-number summary, and construct a boxplot.

Key Terms and Steps:

• Five-number summary: Minimum, Q1 (first quartile), Median, Q3 (third quartile), Maximum

• Boxplot: A graphical representation of the five-number summary

Step-by-Step Guidance

1. List all data values from the stem-and-leaf plot in order.

2. Find the minimum and maximum values.

3. Calculate the median (middle value), Q1 (median of lower half), and Q3 (median of upper half).

4. Draw a number line and mark the five-number summary values.

5. Draw the box from Q1 to Q3, with a line at the median, and whiskers to the minimum and maximum.

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