Unit 1 Statistics Practice Problems

Flashcards covering key statistical concepts, practice problems, sampling techniques, measures of center, experimental design, histograms, and z-score calculations.

In a study of $33,173$ infants in Bangladesh conducted to investigate a link between heart rhythm abnormalities and sudden infant death syndrome (SIDS), what are the population of interest and the sample?

Population: All infants in Bangladesh. Sample: The $33,173$ infants included in the study.

In a 2018 survey of $1202$ households in the U.S. finding that $74\%$ have at least one TV connected to a streaming service, what are the population and the sample?

Population: All households in the US. Sample: The $1202$ households surveyed.

Is the following statement an example of a parameter or a statistic? "The average annual salary for $36$ of a company's accountants is \$\text{68,000}$."

Statistic, because the average given represents only $36$ of the company's accountants rather than all of them.

Is the following statement an example of a parameter or a statistic? "Two percent of the $5147$ fatal injuries in the U.S. in 2017 were due to fires and explosions."

Parameter, because the $2\%$ was obtained from all $5147$ fatal injuries in the US in 2017.

Classify each of the following data types as categorical or quantitative: (a) Street names in Raleigh, (b) Cholesterol level in $\text{mg/dL}$, (c) Answers to "Do you believe climate change is human-caused?", (d) Snow accumulation per day in Minneapolis during January.

(a) Categorical, (b) Quantitative, (c) Categorical, (d) Quantitative.

What design problems exist in a study testing smoking cessation gum where $8$ heavy smokers receive the gum and $8$ receive a placebo, resulting in all $8$ gum users quitting after two months?

It is unknown whether participants were randomly assigned (lack of specified randomization method) and whether participants were aware of receiving treatment or placebo (lack of blinding).

Why is blocking by gender useless in a smoking cessation study where all $1200$ female subjects are given the treatment gum and all male subjects are given a placebo?

Because all females received the gum and all males received the placebo, it is impossible to determine whether the gum works better for women specifically or helps all people quit smoking.

Identify the sampling method: Pungo County is divided into three socioeconomic levels (low income, middle income, high income), and a proportional number of households are randomly chosen from each level.

Stratified sampling, because some households are selected from every group.

Identify the sampling method: Pungo County is divided into subgroups by ZIP code, $10$ ZIP codes are randomly chosen, and all households in those chosen areas are included in the sample.

Cluster sampling, because all households in some selected groups are included.

Identify the sampling method: Among all patients in the UNC Health Care system, $1470$ are randomly chosen to receive a satisfaction survey.

Simple Random sampling, because all patients have the same chance of being selected.

How is the class width determined for a frequency distribution of $18$ burro weights ranging from $61\,kg$ to $183\,kg$ divided into $5$ classes?

Class width = $\frac{183 - 61}{5} = 24.4$, which rounds up to $25$.

Based on the frequency distribution of $18$ burro weights, what are the sample mean and sample standard deviation?

Mean: $\bar{x} = 113.44\,kg$; Standard deviation: $s = 35.89\,kg$. These values are statistics because they are derived from a sample.

What is the Five Number Summary for the dataset of $18$ burro weights, and which measure of center best describes a typical value?

Min = $61\,kg$, $Q_1 = 85\,kg$, Median = $108\,kg$, $Q_3 = 138\,kg$, Max = $183\,kg$. The median best describes a typical value because the dataset is skewed right.

Based on the House Fly Life Span histogram, how many house flies live at least $10$ days?

$20$ house flies ($11 + 4 + 5 = 20$).

Based on the House Fly Life Span histogram, what percentage of house flies live at most $7$ days?

$25\%$ of house flies ($\frac{3 + 7}{3 + 7 + 10 + 11 + 4 + 5} = \frac{10}{40} = 0.25$).

Why is Cuthbert's conclusion that $94\%$ of Americans think stem cell research is murder invalid after surveying people outside a restaurant near a church on a Sunday?

Cuthbert used convenience sampling outside a church, which produces a non-representative sample of the general American population.

At a firm with $30$ engineers earning \$\text{70,000}$, $5$ interns earning \$\text{12,000}$, and $1$ owner earning over \$\text{1,000,000}$, which measure of center best represents typical worker earnings?

The median, because the president's salary of over one million dollars skews the distribution to the right, inflating the mean.

If men's heights follow a bell-shaped distribution with a mean of $176\,cm$ and a standard deviation of $7\,cm$, use the Empirical Rule to find the percentages of men between (a) $169\,cm$ and $183\,cm$, and (b) $155\,cm$ and $197\,cm$.

(a) $68\%$ (within $1$ standard deviation: $176 \pm 7\,cm$), (b) $99.7\%$ (within $3$ standard deviations: $176 \pm 3 \times 7\,cm$).

Human body temperatures follow a bell-shaped distribution with mean $98.2^\circ\text{F}$ and standard deviation $0.62^\circ\text{F}$. Convert $97.5^\circ\text{F}$, $99.6^\circ\text{F}$, and $98.2^\circ\text{F}$ to z-scores and state if any are statistically significant.

$z(97.5) = \frac{97.5 - 98.2}{0.62} = -1.129$; $z(99.6) = \frac{99.6 - 98.2}{0.62} = 2.258$; $z(98.2) = \frac{98.2 - 98.2}{0.62} = 0$. A temperature of $99.6^\circ\text{F}$ is statistically significant (unusual) because its z-score is greater than $2$.

What do z-scores measure, and what do positive versus negative z-scores indicate?

Z-scores measure how many standard deviations a data value lies from the mean, standardizing values across different distributions. A positive z-score lies above the mean, while a negative z-score lies below the mean.

Comparing biathlon winner Magnar Solberg (age $35$, mean $27.3$, SD $4.7$) and speed skating winner Christina Baas-Kaiser (age $33$, mean $23.1$, SD $3.9$), who was relatively older compared to their sport?

Christina was relatively older because her z-score ($z = \frac{33 - 23.1}{3.9} = 2.538$) was higher than Magnar's z-score ($z = \frac{35 - 27.3}{4.7} = 1.638$).

In home sales statistics, why would the mean home price (\$\text{264,000}$) be greater than the median home price (\$\text{221,000}$)?

The mean price is higher than the median because a few homes sold at very high prices, skewing the distribution to the right while the majority of homes sold in a lower price range.

In Laura's study on tomato yield using $3$ varieties of tomato plants in sunny and shaded garden locations, what type of study design was used and how were the pairs constructed?

It was an experiment using a matched pairs design. Two plants of each variety were paired, with one assigned to morning watering (sunny area) and the other to evening watering (shaded area).

Based on the textbook spending histograms, which group of students appears to typically spend more, NCSU or Wake Tech, and which summary is best for Wake Tech's distribution?

NCSU students typically spend more because the center of their distribution is higher ($300\text{--}399$ range). The Five Number Summary is best for Wake Tech because its distribution is right-skewed.

In the journalist's police patrol study, how is the variable "Number of times patrolled" classified?

It is classified as Quantitative and Discrete.