The Normal Distribution, Revisited - AP Statistics
Card 1 of 30
Given $z = 2$, what percentage of data is below this z-score?
Given $z = 2$, what percentage of data is below this z-score?
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Approximately 97.5%. Two standard deviations above mean in standard normal distribution.
Approximately 97.5%. Two standard deviations above mean in standard normal distribution.
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What proportion of data falls within 3 standard deviations in a normal distribution?
What proportion of data falls within 3 standard deviations in a normal distribution?
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Approximately 99.7%. From the empirical rule for normal distributions.
Approximately 99.7%. From the empirical rule for normal distributions.
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What proportion of data falls within 2 standard deviations in a normal distribution?
What proportion of data falls within 2 standard deviations in a normal distribution?
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Approximately 95%. From the empirical rule for normal distributions.
Approximately 95%. From the empirical rule for normal distributions.
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What proportion of data falls within 1 standard deviation in a normal distribution?
What proportion of data falls within 1 standard deviation in a normal distribution?
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Approximately 68%. From the empirical rule for normal distributions.
Approximately 68%. From the empirical rule for normal distributions.
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What does a z-score represent?
What does a z-score represent?
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Number of standard deviations from the mean. Measures distance from mean in standard deviation units.
Number of standard deviations from the mean. Measures distance from mean in standard deviation units.
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State the empirical rule for a normal distribution.
State the empirical rule for a normal distribution.
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68%-95%-99.7% of data within 1, 2, 3 standard deviations. Also known as the 68-95-99.7 rule for normal distributions.
68%-95%-99.7% of data within 1, 2, 3 standard deviations. Also known as the 68-95-99.7 rule for normal distributions.
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Identify the symbol for the mean in a normal distribution.
Identify the symbol for the mean in a normal distribution.
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$\mu$ (mu). Greek letter representing population mean parameter.
$\mu$ (mu). Greek letter representing population mean parameter.
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What is the z-score formula?
What is the z-score formula?
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$z = \frac{x - \mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.
$z = \frac{x - \mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.
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What is the probability of $z < -3$?
What is the probability of $z < -3$?
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Approximately 0.13%. Lower tail probability beyond three standard deviations.
Approximately 0.13%. Lower tail probability beyond three standard deviations.
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What is the probability of $z < -1.96$ or $z > 1.96$?
What is the probability of $z < -1.96$ or $z > 1.96$?
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Approximately 5%. Combined tail probabilities beyond critical values.
Approximately 5%. Combined tail probabilities beyond critical values.
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Find the z-score for the 75th percentile.
Find the z-score for the 75th percentile.
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$z \approx 0.675$. Third quartile z-score in standard normal distribution.
$z \approx 0.675$. Third quartile z-score in standard normal distribution.
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What percentage of data is beyond $z = 3$?
What percentage of data is beyond $z = 3$?
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Approximately 0.13%. Combined tail areas beyond three standard deviations.
Approximately 0.13%. Combined tail areas beyond three standard deviations.
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What is the probability of $z < 0.33$?
What is the probability of $z < 0.33$?
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Approximately 62.97%. Cumulative probability up to 0.33 standard deviations.
Approximately 62.97%. Cumulative probability up to 0.33 standard deviations.
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Find the z-score for $x = 92$, $\mu = 100$, $\sigma = 8$.
Find the z-score for $x = 92$, $\mu = 100$, $\sigma = 8$.
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$z = -1$. Using $z = \frac{92-100}{8} = -1$.
$z = -1$. Using $z = \frac{92-100}{8} = -1$.
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What is the complementary probability for $z > 1.645$?
What is the complementary probability for $z > 1.645$?
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5%. Upper tail probability beyond 1.645 standard deviations.
5%. Upper tail probability beyond 1.645 standard deviations.
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Find the probability of $z = 0.84$.
Find the probability of $z = 0.84$.
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Approximately 79.93%. Cumulative probability up to 0.84 standard deviations.
Approximately 79.93%. Cumulative probability up to 0.84 standard deviations.
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Calculate the probability for $1 < z < 2$.
Calculate the probability for $1 < z < 2$.
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Approximately 13.59%. Area between one and two standard deviations above mean.
Approximately 13.59%. Area between one and two standard deviations above mean.
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What does the term 'standard normal distribution' mean?
What does the term 'standard normal distribution' mean?
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Normal distribution with $\mu = 0$, $\sigma = 1$. Normalized version with mean 0 and standard deviation 1.
Normal distribution with $\mu = 0$, $\sigma = 1$. Normalized version with mean 0 and standard deviation 1.
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Determine the probability of $z < -2$.
Determine the probability of $z < -2$.
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Approximately 2.28%. Lower tail probability beyond two standard deviations.
Approximately 2.28%. Lower tail probability beyond two standard deviations.
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What z-score corresponds to the 90th percentile?
What z-score corresponds to the 90th percentile?
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$z \approx 1.28$. Critical value for 90% of data below this point.
$z \approx 1.28$. Critical value for 90% of data below this point.
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Determine the interquartile range (IQR) for a standard normal distribution.
Determine the interquartile range (IQR) for a standard normal distribution.
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IQR = $1.349$. Distance between 75th and 25th percentiles.
IQR = $1.349$. Distance between 75th and 25th percentiles.
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What is the relationship between mean, median, and mode in a normal distribution?
What is the relationship between mean, median, and mode in a normal distribution?
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They are equal. Symmetry property of normal distributions.
They are equal. Symmetry property of normal distributions.
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Find the z-score for the 25th percentile.
Find the z-score for the 25th percentile.
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$z \approx -0.675$. First quartile z-score in standard normal distribution.
$z \approx -0.675$. First quartile z-score in standard normal distribution.
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Describe the tails of a normal distribution.
Describe the tails of a normal distribution.
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The tails are asymptotic to the x-axis. Approach but never touch the horizontal axis.
The tails are asymptotic to the x-axis. Approach but never touch the horizontal axis.
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What is the cumulative distribution function (CDF) at $z = 0$?
What is the cumulative distribution function (CDF) at $z = 0$?
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0.5. At mean, exactly half the data is below.
0.5. At mean, exactly half the data is below.
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Calculate the probability of $z > 1.96$.
Calculate the probability of $z > 1.96$.
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Approximately 2.5%. Upper tail beyond critical value for 95% confidence.
Approximately 2.5%. Upper tail beyond critical value for 95% confidence.
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Convert $z = 0.5$ to a percentage of data below.
Convert $z = 0.5$ to a percentage of data below.
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Approximately 69.15%. Half standard deviation above mean corresponds to this percentile.
Approximately 69.15%. Half standard deviation above mean corresponds to this percentile.
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Identify the symmetry property of a normal distribution.
Identify the symmetry property of a normal distribution.
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Symmetric around the mean. Mean divides distribution into two equal halves.
Symmetric around the mean. Mean divides distribution into two equal halves.
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Find the x-value for $z = -1$, $\mu = 100$, $\sigma = 10$.
Find the x-value for $z = -1$, $\mu = 100$, $\sigma = 10$.
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$x = 90$. Solving $-1 = \frac{x-100}{10}$ gives $x = 90$.
$x = 90$. Solving $-1 = \frac{x-100}{10}$ gives $x = 90$.
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Identify the symbol for standard deviation in a normal distribution.
Identify the symbol for standard deviation in a normal distribution.
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$\sigma$ (sigma). Greek letter representing population standard deviation parameter.
$\sigma$ (sigma). Greek letter representing population standard deviation parameter.
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