The grades on a statistics test are normally distributed with a mean of 62 and Q1=52. If the instructor wishes to assign B's or higher to the top 30% of the students in the class, what grade is required to get a B or higher? Please round your answer to two decimal places.

Answer:To get a B or higher, you need to get a grade of at least 69.77.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]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. Subtracting 1 by the pvalue, we This p-value is the probability that the value of the measure is greater than X.In this problem, we have thatMean of 62, so [tex]\mu = 62[/tex]Q1 of 52 means that the z score of X = 52 has a pvalue of 0.25. Z has a pvalue of 0.25 between -0.67 and -0.68, so we use [tex]Z = -0.675[/tex][tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]-0.675 = \frac{52 - 62}{\sigma}[/tex][tex]-0.675\sigma = -10[/tex]Multiplying the equality by (-1), we have that:[tex]0.675\sigma = 10[/tex][tex]\sigma = \frac{10}{0.675}[/tex][tex]\sigma = 14.8[/tex]If the instructor wishes to assign B's or higher to the top 30% of the students in the class, what grade is required to get a B or higher?Those are the Z scores that have a pvalue higher than 0.70. So we have to find X when Z has a pvalue of 0.70. This is between 0.52 and 0.53, so we use [tex]Z = 0.525[/tex][tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]0.525 = \frac{X - 62}{14.8}[/tex][tex]X - 62 = 14.8*0.525[/tex][tex]X = 69.77[/tex]To get a B or higher, you need to get a grade of at least 69.77.