Question

If scores are normally distributed with a mean of 35 and a standard deviation of \(10,\) what percent of the scores is: a. greater than 34 ? b. smaller than \(42 ?\) c. between 28 and \(34 ?\)

Short answer

a. 53.98%, b. 75.80%, c. 21.82%

Step-by-step solution

Understanding the Normal Distribution

The problem states that scores are normally distributed with a mean (\( \mu \)) of 35 and a standard deviation (\( \sigma \)) of 10. This means the scores follow the bell-shaped curve, centered at 35, with most scores lying within a few standard deviations from the mean.

Step 2a: Convert Raw Score to z-score (Greater than 34)

To find the percentage of scores greater than 34, calculate the z-score using the formula \( z = \frac{x - \mu}{\sigma} \), where \( x = 34 \), \( \mu = 35 \), and \( \sigma = 10 \). \[ z = \frac{34 - 35}{10} = \frac{-1}{10} = -0.1 \]

Step 3a: Use z-table for Greater than 34

Using a standard normal distribution table (z-table), find the probability associated with \( z = -0.1 \). The table gives the probability that a score is less than 34. For a z-score of -0.1, this is approximately 0.4602. Thus, the probability of being greater than 34 is:\[ P(X > 34) = 1 - 0.4602 = 0.5398 \] or 53.98%.

Step 2b: Convert Raw Score to z-score (Smaller than 42)

For scores smaller than 42, calculate the z-score using \( x = 42 \),\[ z = \frac{42 - 35}{10} = \frac{7}{10} = 0.7 \]

Step 3b: Use z-table for Smaller than 42

Using the z-table, find the probability for \( z = 0.7 \). This gives the probability of a score being less than 42. For a z-score of 0.7, the table shows approximately 0.7580. \[ P(X < 42) = 0.7580 \] or 75.80%.

Step 2c: Convert Raw Scores to z-scores (Between 28 and 34)

Find the z-scores for 28 and 34. For 28:\[ z_{28} = \frac{28 - 35}{10} = -0.7 \]For 34:\[ z_{34} = \frac{34 - 35}{10} = -0.1 \]

Step 3c: Use z-table for Range 28 to 34

Find the probability for each z-score from the z-table. For \( z = -0.7 \), the table shows approximately 0.2420, and for \( z = -0.1 \), it shows approximately 0.4602.The probability that a score is between 28 and 34 is:\[ P(28 < X < 34) = 0.4602 - 0.2420 = 0.2182 \] or 21.82%.

Key concepts

Understanding z-score

The concept of a z-score is fundamental when working with normal distributions. A z-score, often referred to as a standard score, is a way of describing a data point's position relative to the mean of a group of points. The formula to calculate a z-score is \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the data point, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation. By converting scores into z-scores, we can determine how many standard deviations away a particular score is from the mean.

• If a z-score is 0, the data point's score is exactly average or equal to the mean.
• Positive z-scores indicate the score is above the mean, while negative z-scores show it's below the mean.
• This normalization is crucial for comparing scores from different distributions or calculating probabilities.

Understanding z-scores allows you to make sense of how extreme or typical a certain score is within a distribution.

Utilizing the z-table for standard normal distribution

The z-table, also known as the standard normal distribution table, is used to find the probability of a statistic (z-score) falling below or above a certain point in a standard normal distribution.Each z-score corresponds to a unique probability in the z-table. By convention:

• The table shows probabilities for z-scores less than or equal to a given value.
• For probabilities greater than a z-score, subtract the table value from 1.
• Rows and columns intersect to provide the cumulative probability from the left of the standard normal curve up to the z-score.

Using the z-table is a step-by-step process:

• Convert the raw score to a z-score using \( z = \frac{x - \mu}{\sigma} \).
• Locate the corresponding cumulative probability for the z-score in the table.
• For two z-scores, calculate areas for both and subtract to find inter-quartile probability.

This process is the bedrock for calculating probabilities in normally distributed data.

Probability calculation in normal distribution

Calculating probability involves using z-scores and the z-table to understand the likelihood of a particular score range within a normal distribution. Here’s a simple breakdown: - **Calculate the z-score:** Start by converting your raw data point into a z-score, which we learned earlier. This tells you how far and in which direction a data point deviates from the mean in terms of standard deviations. - **Look up the probability in the z-table:** Once you have a z-score, find the corresponding probability in the z-table. The z-table gives you cumulative probabilities, which is the probability that a random variable is less than or equal to a given z-score. - **Interpreting the probability:**

• If you're interested in the probability that the score is greater than a specific z-score, subtract the z-table value from 1.
• To find the probability between two z-scores, subtract the probability of the lower z-score from the higher one.
• The subtraction accounts for the area under the curve, which represents the probability.

Understanding these calculations is crucial for quantifying uncertainty and making predictions based on normal distribution.