According to PrepScholar, the mean SAT test score is 1051 with a standard deviation of 211. Assume the data set for SAT test scores has a symmetrical bell-shaped distribution. A random sample of 60 students was chosen. Answer the following questions:

1. The Z-score that corresponds to the sample mean SAT test score of 986 is _____. (Round to four decimal places.)

2. The Z-score that corresponds to the sample mean SAT test score of 1050 is _. (Round to four decimal places.)

3. The proportion of the sample mean SAT test scores between 986 and 1050 is _______. (Round to two decimal places.)

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To calculate the Z-score corresponding to a sample mean SAT test score of 986, we use the formula: Z = (x bar - μ) / (σ / √n), where x bar is the sample mean (986 in this case), μ is the population mean (1051), σ is the population standard deviation (211), and n is the sample size (60). Plugging in these values, we get Z = (986 - 1051) / (211 / √60) = -0.9147 (rounded to four decimal places).

Similarly, we can calculate the Z-score for a sample mean SAT test score of 1050 using the same formula. Plugging in the values, we get Z = (1050 - 1051) / (211 / √60) = 0.0443 (rounded to four decimal places).

To find the proportion of sample mean SAT test scores between 986 and 1050, we need to calculate the area under the standard normal curve between these two Z-scores. This can be done using a Z-table or statistical software. The area represents the proportion of values within that range. Subtracting the cumulative probability corresponding to the Z-score of 986 from the cumulative probability corresponding to the Z-score of 1050 will give us the desired proportion.

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