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Setler [38]
3 years ago
14

A professor at a local university noted that the grades of her students were normally distributed with a mean of 78 and a standa

rd deviation of 10. The professor has informed us that 16.6 percent of her students received grades of A. What is the minimum score needed to receive a grade of A?
Mathematics
1 answer:
dimulka [17.4K]3 years ago
4 0

Answer:

The minimum score needed to receive a grade of A is 87.74.

Step-by-step explanation:

We are given that a professor at a local university noted that the grades of her students were normally distributed with a mean of 78 and a standard deviation of 10.

The professor has informed us that 16.6 percent of her students received grades of A.

<u><em /></u>

<u><em>Let X = grades of the students</em></u>

SO, X ~ Normal(\mu=78,\sigma^{2} =10^{2})

The z-score probability distribution for normal distribution is given by;

                               Z = \frac{X-\mu}{\sigma} ~ N(0,1)

where, \mu = mean grades = 78

            \sigma = standard deviation = 10

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.

<u>Now, we are given that the professor has informed us that 16.6 percent of her students received grades of A, so the minimum score needed to receive grade A is given by;</u>

       P(X \geq x) = 0.166   {where x is the required minimum score needed}

       P( \frac{X-\mu}{\sigma} \geq \frac{x-78}{10} ) = 0.166

        P(Z \geq \frac{x-78}{10} ) = 0.166

<em>So, the critical value of x in the z table which represents the top 16.6% of the area is given as 0.9741, that is;</em>

<em>                   </em>      \frac{x-78}{10} =0.9741

                         {x-78}{} =0.9741\times 10

<em>                           </em>x = 78 + 9.741 = <u>87.74</u>

Hence, the minimum score needed to receive a grade of A is 87.74.

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