Introducing Statistics: Why Be Normal - AP Statistics
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What is the cumulative distribution function (CDF) used for?
What is the cumulative distribution function (CDF) used for?
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To find probabilities. CDF gives probability that random variable is less than or equal to x.
To find probabilities. CDF gives probability that random variable is less than or equal to x.
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What is the effect of increasing standard deviation on a normal curve?
What is the effect of increasing standard deviation on a normal curve?
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Flattens and widens the curve. Larger σ spreads data more widely around the mean.
Flattens and widens the curve. Larger σ spreads data more widely around the mean.
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What is the effect of decreasing standard deviation on a normal curve?
What is the effect of decreasing standard deviation on a normal curve?
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Narrows and steepens the curve. Smaller σ concentrates data more tightly around the mean.
Narrows and steepens the curve. Smaller σ concentrates data more tightly around the mean.
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Find the probability of $z > 2$ in a standard normal distribution.
Find the probability of $z > 2$ in a standard normal distribution.
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0.0228. Using $1 - 0.9772 = 0.0228$ from standard normal table.
0.0228. Using $1 - 0.9772 = 0.0228$ from standard normal table.
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Calculate the z-score for $x = 10$, $\text{μ} = 8$, $\text{σ} = 1.5$.
Calculate the z-score for $x = 10$, $\text{μ} = 8$, $\text{σ} = 1.5$.
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1.33. Using $z = \frac{10-8}{1.5} = 1.33$.
1.33. Using $z = \frac{10-8}{1.5} = 1.33$.
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Identify the range of the majority of data points in a normal distribution.
Identify the range of the majority of data points in a normal distribution.
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Within $\mu \pm 3\sigma$. Three standard deviations captures 99.7% of all data points.
Within $\mu \pm 3\sigma$. Three standard deviations captures 99.7% of all data points.
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Find the area between $z = -1$ and $z = 1$ in a standard normal distribution.
Find the area between $z = -1$ and $z = 1$ in a standard normal distribution.
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0.6826. This represents the middle 68% of the distribution.
0.6826. This represents the middle 68% of the distribution.
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Which z-score corresponds to the 95th percentile?
Which z-score corresponds to the 95th percentile?
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1.645. Critical z-value that leaves 5% in the upper tail.
1.645. Critical z-value that leaves 5% in the upper tail.
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Find the z-score for $x = 15$, given $\text{μ} = 15$ and $\text{σ} = 5$.
Find the z-score for $x = 15$, given $\text{μ} = 15$ and $\text{σ} = 5$.
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- Using $z = \frac{15-15}{5} = 0$.
- Using $z = \frac{15-15}{5} = 0$.
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What is the probability that a z-score is exactly 0 in a continuous normal distribution?
What is the probability that a z-score is exactly 0 in a continuous normal distribution?
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- Continuous distributions assign zero probability to any single point.
- Continuous distributions assign zero probability to any single point.
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Identify the skewness of a normal distribution.
Identify the skewness of a normal distribution.
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Zero. Perfect symmetry means no skewness in either direction.
Zero. Perfect symmetry means no skewness in either direction.
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Find the area between $z = -2$ and $z = 2$ in a standard normal distribution.
Find the area between $z = -2$ and $z = 2$ in a standard normal distribution.
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0.9545. This represents 95% of the distribution within two standard deviations.
0.9545. This represents 95% of the distribution within two standard deviations.
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Find the probability of $z < -2$ in a standard normal distribution.
Find the probability of $z < -2$ in a standard normal distribution.
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0.0228. Standard normal table gives area left of $z = -2$.
0.0228. Standard normal table gives area left of $z = -2$.
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Which z-score corresponds to the 5th percentile?
Which z-score corresponds to the 5th percentile?
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-1.645. Critical z-value that leaves 5% in the lower tail.
-1.645. Critical z-value that leaves 5% in the lower tail.
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What does a z-score of 0 indicate?
What does a z-score of 0 indicate?
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Value equals the mean. Zero z-score means the observation equals the population mean.
Value equals the mean. Zero z-score means the observation equals the population mean.
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Find the z-score for $x = 7$, $\text{μ} = 5$, $\text{σ} = 1$.
Find the z-score for $x = 7$, $\text{μ} = 5$, $\text{σ} = 1$.
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- Using $z = \frac{7-5}{1} = 2$.
- Using $z = \frac{7-5}{1} = 2$.
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Find the area to the right of $z = -1$ in a standard normal distribution.
Find the area to the right of $z = -1$ in a standard normal distribution.
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0.8413. Area to the right of $z = -1$ equals area to the left of $z = 1$.
0.8413. Area to the right of $z = -1$ equals area to the left of $z = 1$.
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Find the z-score for $x = 12$, given $\text{μ} = 10$ and $\text{σ} = 4$.
Find the z-score for $x = 12$, given $\text{μ} = 10$ and $\text{σ} = 4$.
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0.5. Using $z = \frac{12-10}{4} = 0.5$.
0.5. Using $z = \frac{12-10}{4} = 0.5$.
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What percentage of data falls within one standard deviation in a normal distribution?
What percentage of data falls within one standard deviation in a normal distribution?
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68%. First part of the 68-95-99.7 empirical rule.
68%. First part of the 68-95-99.7 empirical rule.
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What is the kurtosis of a normal distribution?
What is the kurtosis of a normal distribution?
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- Mesokurtic distribution with standard baseline kurtosis value.
- Mesokurtic distribution with standard baseline kurtosis value.
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Find the area to the left of $z = -2$ in a standard normal distribution.
Find the area to the left of $z = -2$ in a standard normal distribution.
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0.0228. Standard normal table gives area left of $z = -2$.
0.0228. Standard normal table gives area left of $z = -2$.
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Identify the empirical rule for normal distributions.
Identify the empirical rule for normal distributions.
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68-95-99.7 rule. Also known as the empirical rule for normal distribution percentages.
68-95-99.7 rule. Also known as the empirical rule for normal distribution percentages.
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What is the key characteristic of a normal distribution's tails?
What is the key characteristic of a normal distribution's tails?
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Asymptotic. Tails approach but never touch the horizontal axis.
Asymptotic. Tails approach but never touch the horizontal axis.
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What is the probability that a z-score is exactly 0 in a continuous normal distribution?
What is the probability that a z-score is exactly 0 in a continuous normal distribution?
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- Continuous distributions assign zero probability to any single point.
- Continuous distributions assign zero probability to any single point.
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State the property of symmetry in a normal distribution.
State the property of symmetry in a normal distribution.
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Symmetrical about the mean. Equal areas on both sides of the mean create perfect symmetry.
Symmetrical about the mean. Equal areas on both sides of the mean create perfect symmetry.
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What is the standard deviation of a standard normal distribution?
What is the standard deviation of a standard normal distribution?
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- Standard normal distribution has unit variance and standard deviation.
- Standard normal distribution has unit variance and standard deviation.
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Identify the empirical rule for normal distributions.
Identify the empirical rule for normal distributions.
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68-95-99.7 rule. Also known as the empirical rule for normal distribution percentages.
68-95-99.7 rule. Also known as the empirical rule for normal distribution percentages.
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What percentage of data falls within two standard deviations in a normal distribution?
What percentage of data falls within two standard deviations in a normal distribution?
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95%. Second part of the 68-95-99.7 empirical rule.
95%. Second part of the 68-95-99.7 empirical rule.
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What percentage of data falls within three standard deviations in a normal distribution?
What percentage of data falls within three standard deviations in a normal distribution?
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99.7%. Third part of the 68-95-99.7 empirical rule.
99.7%. Third part of the 68-95-99.7 empirical rule.
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What is the notation for a normal distribution with mean $\text{μ}$ and standard deviation $\text{σ}$?
What is the notation for a normal distribution with mean $\text{μ}$ and standard deviation $\text{σ}$?
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$N(\text{μ}, \text{σ})$. Standard notation where μ is mean and σ is standard deviation.
$N(\text{μ}, \text{σ})$. Standard notation where μ is mean and σ is standard deviation.
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