1.10 The Normal Distribution

• In a normal distribution, the mean, median, and mode are all equal
• The z-score is a standardized measure of how far a data point is from the mean
• The normal distribution is perfectly symmetric around its mean.


  True
• Approximately 68% of data in a normal distribution falls within 1 standard deviation of the mean.


  True
• Steps to find the probability using a z-score

  1️⃣ Calculate the z-score using the formula

  2️⃣ Look up the z-score in a standard normal probability table

  3️⃣ Find the corresponding probability
• What is the formula for calculating the z-score?

  z = \frac{x - \mu}{\sigma}</latex>
• What is the probability associated with a z-score of 1 in a standard normal distribution?

  0.8413
• The normal distribution is perfectly symmetric around its mean.

  True
• Approximately 68% of data falls within 1 standard deviation of the mean in a normal distribution.


  True
• A standard normal probability table provides probabilities for a normal distribution with a mean of 0 and a standard deviation of 1.


  True
• What shape characterizes the normal distribution?

  Bell-shaped
• Match the distance from the mean with the approximate percentage of data in a normal distribution:

  Within 1 $\sigma$ ↔️ 68%

  Within 2 $\sigma$ ↔️ 95%

  Within 3 $\sigma$ ↔️ 99.7%
• The z-score formula is $z =$$\frac{x - \mu}{\sigma}$, where \sigma</latex> represents the standard deviation
• A standard normal probability table provides the probability of a data point falling to the right of a z-score.

  False
• For a normal distribution with $\mu =$$50$ and $\sigma =$$10$, the z-score for x = 60</latex> is 1.


  True
• The z-score is calculated using the mean ($\mu$) and the standard deviation
• What are the two parameters that characterize the normal distribution?

  $\mu$ and $\sigma$
• Steps in using the empirical rule to estimate data spread

  1️⃣ Identify the mean ($\mu$) and standard deviation ($\sigma$)

  2️⃣ Calculate the intervals within 1, 2, and 3 $\sigma$ of the mean

  3️⃣ Apply the 68-95-99.7 rule to estimate data within each interval
• What is the formula for calculating the z-score?

  z = \frac{x - \mu}{\sigma}</latex>
• Match the z-score with its corresponding probability in a standard normal distribution:

  1 ↔️ 0.8413

  2 ↔️ 0.9772

  -1 ↔️ 0.1587
• The parameters of the normal distribution are the mean ($\mu$) and the standard deviation
• What does the z-score measure?

  Distance from the mean
• What is the probability associated with a z-score of 1 in a standard normal distribution?

  0.8413
• A standard normal probability table is used to find the probability of a data point falling to the left of a particular z-score