Statistics Flashcards: Using Normal Distributions To Estimate Populations

What this deck covers

This deck focuses on Using Normal Distributions To Estimate Populations, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Using the Empirical Rule, estimate $P(X<\mu-3\sigma)$ for normal data.

Answer: About $0.0015$. 99.7% within $3\sigma$, so 0.3% in both tails, 0.15% in lower tail.

Flashcard 2: What is the mean and standard deviation of the standard normal distribution?

Answer: Mean $0$, standard deviation $1$. Standard normal is centered at 0 with unit spread.

Flashcard 3: What does the notation $\Phi(z)$ represent in normal distribution tables?

Answer: $\Phi(z)=P(Z\le z)$ for $Z\sim N(0,1)$. Cumulative distribution function for standard normal.

Flashcard 4: Find $x$ for $z=-2$ in a normal model with $\mu=50$ and $\sigma=4$.

Answer: $x=42$. Apply $x=50+(-2)(4)=50-8$.

Flashcard 5: What percentage of data is within $3\sigma$ of the mean in an approximately normal distribution?

Answer: About $99.7\%$. Third part of the 68-95-99.7 rule for normal distributions.

Flashcard 6: Which condition is required to estimate percentages using a normal model: strong skew, outliers, or roughly bell-shaped?

Answer: Roughly bell-shaped and symmetric with no strong outliers. Normal model requires symmetric, unimodal data.

Flashcard 7: What is the probability expression for being between two $z$-scores $a$ and $b$ in $N(0,1)$?

Answer: $P(a<Z<b)=\Phi(b)-\Phi(a)$. Interval probability equals difference of cumulative probabilities.

Flashcard 8: Which situation makes a normal model inappropriate: unimodal symmetric data or strongly right-skewed data?

Answer: Strongly right-skewed data. Skewed distributions violate normality assumption.

Flashcard 9: What is the 68-95-99.7 rule percentage within $1\sigma$ of the mean for a normal model?

Answer: About $68\%$. First part of the empirical rule for normal distributions.

Flashcard 10: What proportion of a normal distribution lies above the mean $\mu$?

Answer: $0.50$. Normal distribution is symmetric about the mean.

Flashcard 11: Using the Empirical Rule, estimate $P(X>\mu+2\sigma)$ for normal data.

Answer: About $0.025$. 95% within $2\sigma$, so 5% in both tails, 2.5% in upper tail.

Flashcard 12: Which plot is most commonly used to check whether a data set is approximately normal: dotplot, histogram, or boxplot?

Answer: Histogram. Shows distribution shape to assess normality visually.

Flashcard 13: What percentage of data is within $1\sigma$ of the mean in an approximately normal distribution?

Answer: About $68\%$. First part of the 68-95-99.7 rule for normal distributions.

Flashcard 14: What is the empirical rule estimate for the proportion between $\mu$ and $\mu+1\sigma$?

Answer: About $34\%$. Half of $68\%$ lies between mean and one SD above.

Flashcard 15: What is the $z$-score formula for a value $x$ given mean $\mu$ and standard deviation $\sigma$?

Answer: $z=\frac{x-\mu}{\sigma}$. Measures how many standard deviations a value is from the mean.

Flashcard 16: Using a $z$-table where $\Phi(0.80)=0.7881$, what is $P(-0.80<Z<0.80)$?

Answer: $0.5762$. By symmetry: $2(0.7881)-1$ or $0.7881-(1-0.7881)$.

Flashcard 17: What is the $z$-score formula for a data value $x$ with mean $\mu$ and standard deviation $\sigma$?

Answer: $z=\frac{x-\mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.

Flashcard 18: Find $z$ for $x=85$, $\mu=70$, $\sigma=5$.

Answer: $z=3$. $z=\frac{85-70}{5}=\frac{15}{5}=3$

Flashcard 19: What is the 68-95-99.7 rule percentage within $2\sigma$ of the mean for a normal model?

Answer: About $95\%$. Second part of the empirical rule for normal distributions.

Flashcard 20: Using the empirical rule, estimate $P(\mu-1\sigma<X<\mu+1\sigma)$ for a normal model.

Answer: About $68\%$. Direct application of the $68\%$ empirical rule.

Flashcard 21: What is the probability formula for being between two $z$-scores $a$ and $b$ using the normal CDF $\Phi$?

Answer: $P(a<Z<b)=\Phi(b)-\Phi(a)$. Difference of CDFs gives probability between two values.

Flashcard 22: If a table gives $\Phi(1.25)=0.8944$, what is $P(Z>1.25)$?

Answer: $0.1056$. $P(Z>1.25)=1-0.8944=0.1056$

Flashcard 23: What is the formula to convert a $z$-score to a data value $x$ using $\mu$ and $\sigma$?

Answer: $x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.

Flashcard 24: What is the complement rule for a standard normal probability $P(Z>z)$ in terms of $P(Z\le z)$?

Answer: $P(Z>z)=1-P(Z\le z)$. Uses the complement rule for probability.

Flashcard 25: Identify the correct condition: When is fitting a normal model to data generally inappropriate?

Answer: When data are strongly skewed or have outliers. Normal models assume symmetric, bell-shaped data.

Flashcard 26: Identify the distribution model used to fit data with mean $\mu$ and standard deviation $\sigma$ as normal.

Answer: $X\sim N(\mu,\sigma)$. Standard notation for normal distribution with parameters.

Flashcard 27: Using the Empirical Rule, estimate $P(\mu-\sigma<X<\mu+\sigma)$ for normal data.

Answer: About $0.68$. 68% rule: about 68% within one standard deviation.

Flashcard 28: Identify the correct condition: What plot shape most supports using a normal model for a data set?

Answer: A roughly symmetric, unimodal, bell-shaped distribution. These characteristics match the normal distribution shape.

Flashcard 29: Find $z$ for $x=85$ in a normal model with $\mu=70$ and $\sigma=10$.

Answer: $z=1.5$. Apply $z=\frac{85-70}{10}=\frac{15}{10}$.

Flashcard 30: What is the inverse conversion from a $z$-score to a data value $x$ for mean $\mu$ and SD $\sigma$?

Answer: $x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.

Flashcard 31: Find $x$ when $z=-1.5$, $\mu=100$, and $\sigma=20$.

Answer: $x=70$. $x=100+(-1.5)(20)=100-30=70$

Flashcard 32: What symmetry rule relates $P(Z\le -z)$ and $P(Z\le z)$ for $Z\sim N(0,1)$?

Answer: $P(Z\le -z)=1-\Phi(z)$. Uses symmetry of normal curve around mean 0.

Flashcard 33: Using the empirical rule, estimate $P(X>\mu+2\sigma)$ for a normal model.

Answer: About $2.5\%$. $(100\%-95\%)/2$ gives the upper tail beyond $2\sigma$.

Flashcard 34: What is the symmetry rule for normal models relating $P(Z\le z)$ and $P(Z\ge -z)$?

Answer: $P(Z\le z)=P(Z\ge -z)$. Normal distribution is symmetric about zero for standard normal.

Flashcard 35: What is the 68-95-99.7 rule percentage within $3\sigma$ of the mean for a normal model?

Answer: About $99.7\%$. Third part of the empirical rule for normal distributions.

Flashcard 36: What is the complement rule for a standard normal probability such as $P(Z>z)$?

Answer: $P(Z>z)=1-\Phi(z)$. Right tail probability equals 1 minus left tail probability.

Flashcard 37: Using a $z$-table where $\Phi(1.25)=0.8944$, what is $P(Z>1.25)$?

Answer: $0.1056$. Apply complement rule: $1-0.8944$.

Flashcard 38: What percentage of data is within $2\sigma$ of the mean in an approximately normal distribution?

Answer: About $95\%$. Second part of the 68-95-99.7 rule for normal distributions.

Flashcard 39: What is the empirical rule estimate for the proportion between $\mu+1\sigma$ and $\mu+2\sigma$?

Answer: About $13.5\%$. $(95\%-68\%)/2$ gives this tail proportion.

Flashcard 40: What is the empirical rule estimate for the proportion between $\mu+2\sigma$ and $\mu+3\sigma$?

Answer: About $2.35\%$. $(99.7\%-95\%)/2$ gives this tail proportion.