Goodreads is a website where users can rate books they’ve read and share their ratings with friends. Books are rated on a scale from 1 to 5 stars, with larger numbers representing better books. A random selection of books has overall ratings that are bell-shaped with mean 3.7 and standard deviation 0.37. A new, popular book has an average rating of 4.6 on Goodreads. Determine the z-score for this book.

To determine the \( z \)-score for the book with an average rating of 4.6, we use the formula for the \( z \)-score:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \( x \) is the value of interest (the book's rating, 4.6),
- \( \mu \) is the mean of the distribution (3.7),
- \( \sigma \) is the standard deviation (0.37).

Plugging in the values:

\[
z = \frac{4.6 - 3.7}{0.37}
\]

Now let's calculate it:

\[
z = \frac{0.9}{0.37} \approx 2.43
\]

Thus, the \( z \)-score for the book is approximately **2.43**. 

This means that the book's rating is 2.43 standard deviations above the mean.

Would you like further details or have any questions?

Here are five related questions to expand your understanding:
1. What does a positive \( z \)-score indicate about a data point?
2. How would you interpret a \( z \)-score of 0?
3. What percentage of books would you expect to have ratings lower than this book (using the normal distribution)?
4. How would you calculate the probability of a book having a rating higher than 4.6?
5. How does the standard deviation affect the spread of ratings in this context?

**Tip:** A higher absolute value of a \( z \)-score indicates that the data point is further from the mean, either in the positive or negative direction.

Discover how to calculate the z-score for a Goodreads book rating of 4.6, given a mean of 3.7 and a standard deviation of 0.37. This statistical problem is solved using the z-score formula, providing insights into the book's rating relative to the normal distribution. Suitable for Grades 10-12.

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