how to sketch the areas under the standard normal curveover the indicated intervals and find the specified areas
to the left z=0 to the left z=-0.47 to the left of z=0.45
50% is to the left of Z = 0 = mean
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the other Z scores.
To sketch the areas under the standard normal curve and find the specified areas, you can follow these steps:
1. Understand the standard normal distribution: The standard normal distribution, also known as the Z-distribution, has a mean of 0 and a standard deviation of 1. It is a symmetrical bell-shaped curve.
2. Sketch the standard normal curve: On a graph or using software like Excel, plot the standard normal curve. The x-axis represents the Z-scores, and the y-axis represents the probability density.
3. Indicate the intervals: For each specified interval, draw vertical lines on the x-axis to indicate the Z-score values. In this case, you need to sketch the areas to the left of three different Z-scores: 0, -0.47, and 0.45.
4. Shade the areas: To find the specified areas, you need to shade the regions under the curve up to the indicated Z-scores. Shade the area to the left of each Z-score value.
5. Calculate the areas using the standard normal distribution table: To find the specified areas, you can use the standard normal distribution table or software like Excel. Locate the Z-score value on the table and identify the corresponding area under the curve.
- For Z = 0: The area to the left of Z = 0 is 0.5. This means that 50% of the data falls to the left of Z = 0 in a standard normal distribution.
- For Z = -0.47: Refer to the standard normal distribution table to find the area corresponding to Z = -0.47. In the table, the intersection of the row containing -0.4 and the column containing 0.07 gives you the area, which is approximately 0.3192 or 31.92%.
- For Z = 0.45: Locate Z = 0.4 in the row and 0.05 in the column to find the area, which is approximately 0.6736 or 67.36%.
Keep in mind that you can also use statistical software or online calculators to find these areas instead of the standard normal distribution table.