A First Course in Probability (10th Edition)
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ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Transcribed Image Text:Certainly! Below is a transcription of the normal distribution table (also known as the z-table) shown in the image. This table provides cumulative probabilities of the standard normal distribution from negative infinity to z.
### Normal Distribution Table (Z-Table)
The table is organized with `z` scores in the first column and probabilities in the remaining columns. Each cell in the table provides the probability that a standard normal random variable will be less than or equal to the corresponding z-score.
#### Table Structure:
- **z column**:
- Lists the z-scores (standard deviations from the mean) in increments of 0.1, ranging from 0.0 to 3.4.
- **Columns 0.00 to 0.09**:
- Each row under these columns represents the probability for the z-score in that row plus the additional value from the column header.
#### Excerpt from the Table:
| z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| 0.0 | 0.5000| 0.5040| 0.5080| 0.5120| 0.5160| 0.5199| 0.5239| 0.5279| 0.5319| 0.5359|
| 0.1 | 0.5398| 0.5438| 0.5478| 0.5517| 0.5557| 0.5596| 0.5636| 0.5675| 0.5714| 0.5753|
| 0.2 | 0.5793| 0.5832| 0.5871| 0.5910| 0.5948| 0.5987| 0.6026| 0.6064| 0.6103| 0.6141|
| 0.3 | 0.6179| 0.6217| 0.6255|
![**Understanding Probability with the Normal Distribution Table**
The diameters of ball bearings are distributed normally. The mean diameter is 53 millimeters, and the variance is 25. To find the probability that the diameter of a selected bearing is greater than 48 millimeters, you would use the Normal Distribution Table, as provided below. Round your answer to four decimal places.
**Normal Distribution Table Explanation**
The table provided is a standard normal distribution table which shows values for the cumulative distribution function of the standard normal distribution (Z-table). This table is used to determine the probability that a standard normal random variable is less than or equal to a given value.
**How to Use the Table:**
1. **Identify the Z-Score:** Calculate the Z-score for your specific situation. This requires you to subtract the mean from your value of interest, and then divide by the standard deviation. The formula is:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation (which is the square root of the variance).
2. **Find the Probability:** Locate the Z-score in the table to find the probability that a value is less than or equal to your specific value.
3. **Interpreting the Table:**
- The leftmost column represents the Z-value's whole number and tenths place.
- The top row shows the hundredths place of the Z-value.
- At the intersection of your Z-value row and hundredths column, you find the cumulative probability.
4. **Calculate Your Probability:** Since you want the probability greater than a certain value, subtract the table value from 1.
**Normal Table Example:**
- For Z = -3.4, under column 0.09, the value is 0.0002.
- For Z = -3.0, the values range from 0.0013 at column 0.00 to 0.0025 at column 0.09.
- For Z = 0.0, the probabilities range from 0.5000 (at column 0.00) to 0.5359 (at column 0.09).
Remember, this table only shows values for Z-scores from -3.4 to 0.0.
By following these steps and utilizing](https://content.bartleby.com/qna-images/question/96e10c9a-0d86-4b33-b8d4-613a7b6f378e/bb3691c1-d000-4565-a9dd-a93823575b8a/sxvdm1_processed.jpeg)
Transcribed Image Text:**Understanding Probability with the Normal Distribution Table**
The diameters of ball bearings are distributed normally. The mean diameter is 53 millimeters, and the variance is 25. To find the probability that the diameter of a selected bearing is greater than 48 millimeters, you would use the Normal Distribution Table, as provided below. Round your answer to four decimal places.
**Normal Distribution Table Explanation**
The table provided is a standard normal distribution table which shows values for the cumulative distribution function of the standard normal distribution (Z-table). This table is used to determine the probability that a standard normal random variable is less than or equal to a given value.
**How to Use the Table:**
1. **Identify the Z-Score:** Calculate the Z-score for your specific situation. This requires you to subtract the mean from your value of interest, and then divide by the standard deviation. The formula is:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation (which is the square root of the variance).
2. **Find the Probability:** Locate the Z-score in the table to find the probability that a value is less than or equal to your specific value.
3. **Interpreting the Table:**
- The leftmost column represents the Z-value's whole number and tenths place.
- The top row shows the hundredths place of the Z-value.
- At the intersection of your Z-value row and hundredths column, you find the cumulative probability.
4. **Calculate Your Probability:** Since you want the probability greater than a certain value, subtract the table value from 1.
**Normal Table Example:**
- For Z = -3.4, under column 0.09, the value is 0.0002.
- For Z = -3.0, the values range from 0.0013 at column 0.00 to 0.0025 at column 0.09.
- For Z = 0.0, the probabilities range from 0.5000 (at column 0.00) to 0.5359 (at column 0.09).
Remember, this table only shows values for Z-scores from -3.4 to 0.0.
By following these steps and utilizing
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