A population of values has a normal distribution with [tex]\mu = 97.7[/tex] and [tex]\sigma = 96.2[/tex]. You intend to draw a random sample of size [tex]n = 159[/tex].

1. Find the probability that a single randomly selected value is between 85.5 and 122.1. Express your answer as [tex]P(85.5 < X < 122.1)[/tex].

2. Find the probability that a sample of size [tex]n = 159[/tex] is randomly selected with a mean between 85.5 and 122.1. Express your answer as [tex]P(85.5 < \bar{X} < 122.1)[/tex].

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

To find the probability that a single randomly selected value is between 85.5 and 122.1, we can calculate the z-scores corresponding to these values and then use the standard normal distribution table or a statistical software to find the area under the curve between these z-scores.

First, we calculate the z-scores for 85.5 and 122.1 using the formula: z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation.

For 85.5:

z1 = (85.5 - 97.7) / 96.2 = -0.127

For 122.1:

z2 = (122.1 - 97.7) / 96.2 = 0.253

Next, we use the standard normal distribution table or a statistical software to find the area between these z-scores. The area represents the probability of a single randomly selected value falling between 85.5 and 122.1.

Using the standard normal distribution table or a statistical software, we find that the area between z1 = -0.127 and z2 = 0.253 is approximately 0.4638.

Therefore, the probability that a single randomly selected value is between 85.5 and 122.1 is approximately 0.4638.

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