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Part 1

Find the indicated IQ score. The graph to the right depicts IQ scores of​ adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

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xx

0.550.55

A graph with a bell-shaped curve, divided into 2 regions by a line from top to bottom, on the right side. The region left of the line is shaded and labeled 0.55. The x-axis below the line is labeled "x".

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Part 1

The indicated IQ​ score, x, is 

enter your response here. 

​(Round to one decimal place as​ needed.) 

To find the indicated IQ score corresponding to a cumulative area of 0.55 in a normally distributed dataset with a mean of 100 and a standard deviation of 15, you need to follow these steps:

1. Find the z-score corresponding to the cumulative area of 0.55.
2. Convert the z-score to the actual IQ score using the given mean and standard deviation.

### Step 1: Find the z-score
The cumulative area of 0.55 means that 55% of the data lies to the left of the corresponding z-score. Using a z-table or a statistical calculator, we find:

\[ z \approx 0.13 \]

### Step 2: Convert the z-score to the IQ score
Use the following formula to convert the z-score to the actual IQ score:

\[ x = \mu + z \sigma \]

Where:
- \( \mu \) is the mean (100)
- \( \sigma \) is the standard deviation (15)
- \( z \) is the z-score (0.13)

Plugging in the values:

\[ x = 100 + 0.13 \times 15 \]

\[ x = 100 + 1.95 \]

\[ x = 101.95 \]

Rounding to one decimal place:

\[ x \approx 102.0 \]

Therefore, the indicated IQ score is approximately:

\[ x = 102.0 \]

If you have any questions or need further details, feel free to ask!

Here are some related questions you might find helpful:

1. What is the IQ score corresponding to a cumulative area of 0.25?
2. How do you find the IQ score if the area to the right is given instead of the left?
3. What is the area under the normal curve between IQ scores of 85 and 115?
4. How do you interpret an IQ score in a normally distributed dataset?
5. How can you find the IQ score for a given percentile?
6. What is the relationship between z-scores and IQ scores in a normal distribution?
7. How do you use a z-table to find cumulative areas for IQ scores?
8. What is the cumulative area to the left of an IQ score of 100 in a normal distribution?

**Tip:** When converting between z-scores and raw scores in a normal distribution, always remember to use the mean and standard deviation specific to the dataset you are working with.

Learn how to find an IQ score from a normal distribution with a mean of 100 and standard deviation of 15. This guide includes a detailed explanation of using z-scores and the IQ score formula. Ideal for college-level students and those studying statistics.

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