To find the mean ($\mu$) and standard deviation ($\sigma$) of a normal distribution, given that 31% of items are under 45 and 8% are over 64, we can use the properties of the standard normal distribution (Z-distribution).

Step 1: Convert percentages to Z-scores

• The 31% under 45 corresponds to a cumulative probability of 0.31.
• The 8% over 64 corresponds to a cumulative probability of 1 - 0.08 = 0.92.

Using Z-tables (or a statistical calculator), we can find the Z-scores corresponding to these probabilities.

1. For the cumulative probability of 0.31: The Z-score corresponding to 0.31 is approximately -0.50.
2. For the cumulative probability of 0.92: The Z-score corresponding to 0.92 is approximately 1.41.

Step 2: Set up the equations

Using the formula for the Z-score: $Z = \frac{X - \mu}{\sigma}$ where $X$ is the value in the distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation.

For $X = 45$ (31% cumulative): $-0.50 = \frac{45 - \mu}{\sigma} \quad \text{(Equation 1)}$

For $X = 64$ (92% cumulative): $1.41 = \frac{64 - \mu}{\sigma} \quad \text{(Equation 2)}$

Step 3: Solve the equations

We now solve these two simultaneous equations for $\mu$ (mean) and $\sigma$ (standard deviation).

1. From Equation 1: $\mu = 45 + 0.50\sigma$

2. From Equation 2: $\mu = 64 - 1.41\sigma$

Now, we equate the two expressions for $\mu$: $45 + 0.50\sigma = 64 - 1.41\sigma$

Solving for $\sigma$: $1.91\sigma = 64 - 45$ $1.91\sigma = 19$ $\sigma = \frac{19}{1.91} \approx 9.95$

Step 4: Find $\mu$

Substituting $\sigma \approx 9.95$ into Equation 1: $\mu = 45 + 0.50 \times 9.95 \approx 49.98$

Final Answer:

• The mean $\mu$ is approximately 49.98.
• The standard deviation $\sigma$ is approximately 9.95.

Would you like further details on any of these steps, or do you have any questions?

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Here are 5 related questions you might find interesting:

1. How do Z-scores relate to the standard normal distribution?
2. How can you find the cumulative probability using a Z-table?
3. What is the significance of the empirical rule (68-95-99.7 rule) in a normal distribution?
4. How do you calculate probabilities for non-standard normal distributions?
5. What are some real-life examples of normally distributed data?

Tip: When working with normal distributions, always check if the problem involves standard normal (mean = 0, standard deviation = 1) or another form, so you apply the right formulas and Z-scores.