A data set has mean 75 and standard deviation 9. What z-score corresponds to 91
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Z-scores are standardized values that tell us how many standard deviations a data point is from the mean. They allow us to compare data points from different distributions. The z-score formula is z equals x minus mu divided by sigma, where x is the data value, mu is the mean, and sigma is the standard deviation.
In our problem, we're given the mean mu equals 75, the standard deviation sigma equals 9, and we want to find the z-score for the data value x equals 91. Let's visualize these values on a number line to understand their relationship. The mean is at 75, and one standard deviation away from the mean would be at 66 and 84.
Let's visualize our data within a normal distribution with mean 75 and standard deviation 9. The curve shows the probability density of our data. The mean is at 75, and we've marked standard deviation intervals at 66, 75, 84, and 93. Our data point at 91 falls between 84 and 93. The shaded area shows the region between the mean and our data point, representing how far our value is from the center of the distribution.
Now let's calculate the z-score step by step. First, we start with the z-score formula: z equals x minus mu divided by sigma. Next, we substitute our known values: z equals 91 minus 75 divided by 9. Then we perform the subtraction in the numerator: z equals 16 divided by 9. Finally, we complete the division to get z equals 1.78. This means our data point is 1.78 standard deviations above the mean.
Our calculated z-score of 1.78 tells us that the value 91 is 1.78 standard deviations above the mean of 75. In the context of our normal distribution, this places the value in the upper portion of the distribution. The shaded area to the right of our value represents approximately the top 4 percent of the data, meaning our value is relatively high compared to the rest of the dataset.
Let's verify our calculation by working backwards. If z equals 1.78, then x equals mu plus z times sigma. Substituting our values: x equals 75 plus 1.78 times 9. This gives us x equals 75 plus 16.02, which equals 91.02, approximately 91. This confirms our calculation. To summarize the process: first, identify the z-score formula; second, substitute the known values; third, calculate the result; and fourth, interpret what the z-score means. Our final answer is z equals 1.78.