A population of values has a normal distribution with μ = 149.8 μ=149.8 and σ = 68.2 σ=68.2. You intend to draw a random sample of size n = 186 n=186. Please show your answers as numbers accurate to at least 3 decimal places. Find the probability that a single randomly selected value is less than 148.3. Find the probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.

Answer:There is a 49.20% probability that a single randomly selected value is less than 148.3.There is a 38.21% probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.Step-by-step explanation:Problems of normally distributed samples can be solved using the z-score formula.In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by[tex]Z = \frac{X - \mu}{\sigma}[/tex]The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.In this problem, we have that:A population of values has a normal distribution with [tex]\mu = 149.8[/tex] and [tex]σ=68.2[/tex]. Find the probability that a single randomly selected value is less than 148.3This is the pvalue of Z when [tex]X = 148.3[/tex].[tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]Z = \frac{148.3 - 149.8}{68.2}[/tex][tex]Z = -0.02[/tex][tex]Z = -0.02[/tex] has a pvalue of 0.4920There is a 49.20% probability that a single randomly selected value is less than 148.3.Find the probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.We want to find the mean of the sample, so we have to find the standard deviation of the population. That is[tex]s = \frac{68.2}{\sqrt{186}} = 5[/tex]Now, we have to find the pvalue of Z when [tex]X = 148.3[/tex].[tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]Z = \frac{148.3 - 149.8}{5}[/tex][tex]Z = -0.3[/tex][tex]Z = -0.3[/tex] has a pvalue of 0.3821There is a 38.21% probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.