To find the z-scores such that 40% of the data lies between them, we can follow these steps:
1. Identify the desired area: Since we want 40% of the data to lie between the two z-scores, we know that the total area to the left of the first z-score and the total area to the right of the second z-score should be (100% - 40%) / 2 = 30%.
2. Determine the corresponding z-scores: We can use a standard normal distribution table (also known as a z-table) to find the z-scores that correspond to an area of 30% to the left and 30% to the right.
Alternatively, we can use technology (such as statistical software or an online calculator) to find the z-scores. For instance, we could use the inverse normal function on a calculator to find the z-scores.
Note: Since more than one answer is possible, we will find two pairs of z-scores.
Now, let's sketch the distribution and find the corresponding z-scores.
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In the sketch above, the shaded portion represents 40% of the area under the curve.
To find the z-scores, we need to determine the z-score corresponding to 30% to the left and the z-score corresponding to 30% to the right.
Using a z-table or calculator, we find that the z-score corresponding to an area of 30% to the left is approximately -0.524. This means that 30% of the data lies to the left of a z-score of -0.524.
Similarly, the z-score corresponding to an area of 30% to the right is approximately 0.524. This implies that 30% of the data lies to the right of a z-score of 0.524.
Therefore, two pairs of possible z-scores that satisfy this condition are:
1) -0.524 and 0.524
2) -0.524 and -0.524
Keep in mind that z-scores can be positive or negative, so either pair is valid.