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Calculating the Z-score is an essential step when it comes to analyzing normally distributed data. It helps us understand how far away a particular value is from the mean, measured in standard deviations. The formula to calculate the Z-score is quite straightforward: \[ Z = \frac{X - \mu}{\sigma} \] - **\( Z \)** is the Z-score. - **\( X \)** is the value for which you are calculating the Z-score. - **\( \mu \)** is the mean of the distribution. - **\( \sigma \)** is the standard deviation. For example, to find the Z-score for a Douglas fir with a diameter of 3 inches, where the mean diameter \( \mu \) is 4 inches and the standard deviation \( \sigma \) is 1.5 inches, we substitute these values into the formula:\[ Z_1 = \frac{3 - 4}{1.5} = -0.67 \] This tells us that a 3-inch diameter is 0.67 standard deviations below the mean.

The cumulative proportion or cumulative probability is the accumulated probability up to a certain point in a distribution. In normal distribution scenarios, it helps us find the probability that a value is less than or equal to a specific number, using Z-score. Once Z-scores are calculated, they are used to locate corresponding cumulative proportions from a standard normal distribution table (Z-table). Assuming you have a Z-score, you would look this value up in the Z-table to find the cumulative probability. For instance, a Z-score of -0.67 corresponds to a cumulative proportion of approximately 0.2514. This value tells us that about 25.14% of the trees have diameters less than 3 inches. **Steps**:

• Calculate the Z-score for a value.
• Use the Z-table to find the cumulative proportion.
• Interpret the cumulative proportion according to the problem statement.

Understanding these steps can greatly help in solving real-world problems involving normal distributions.

The standard deviation is a key concept in statistics that measures the amount of variation or dispersion of a set of values. In a normal distribution, the standard deviation helps understand how spread out the values are from the mean.A smaller standard deviation indicates that the values tend to be close to the mean, whereas a larger standard deviation means the values are more spread out. For the Douglas fir tree diameters, which are said to be normally distributed:- **Mean (\( \mu \))**: 4 inches- **Standard Deviation (\( \sigma \))**: 1.5 inchesA standard deviation of 1.5 inches suggests that the diameters of the trees vary, creating a spread around the mean of 4 inches. This information is integral when calculating Z-scores and understanding cumulative proportions.

The mean is a central value in a dataset, often referred to as the average. It is calculated by summing all individual data points and dividing by the total number of points. In the context of normally distributed data like the diameters of Christmas trees, the mean indicates the central tendency around which all measurements are clustered. For instance, in the case of Douglas fir trees: - The given mean diameter is 4 inches.
- This means most trees will have diameters close to 4 inches. Knowing the mean helps in calculating Z-scores, enabling us to interpret the data effectively. It forms the basis for various statistical analyses, especially when dealing with the normal distribution. When combined with the standard deviation, the mean gives a complete picture of the data's overall distribution.