Question

After the quizzes are marked, you’re told that the scores on the exam were distributed normally (μ=50, σ=10). You are also told that students with scores in the top 10% will receive a chocolate bar, and students with scores in the top 25% will receive a stick of gum.
a) What is the minimum score needed to receive a chocolate bar?
b)Would a score of 57 earn a stick of gum?

I don't know the steps, so confused!

Answer 1

Answer on Question #49745 – Math – Statistics and Probability

After the quizzes are marked, you're told that the scores on the exam were distributed normally (μ=50, σ=10). You are also told that students with scores in the top 10% will receive a chocolate bar, and students with scores in the top 25% will receive a stick of gum.

a) What is the minimum score needed to receive a chocolate bar?

b) Would a score of 57 earn a stick of gum?

Solution

We need to convert the normal distribution to the standard normal distribution:

$$Z = \frac {x - \mu}{\sigma}.$$

a) $P(X > x_{1}) = P(Z > z_{1}) = 1 - P(Z < z_{1}) = 0.1 \rightarrow P(Z < z_{1}) = 0.9.$

Then from normal table, $z_{1} = 2.33$.

So

$$z _ {1} = \frac {x _ {1} - \mu}{\sigma} \rightarrow x _ {1} = \mu + z _ {1} \sigma = 5 0 + 2. 3 3 \cdot 1 0 = 7 3. 3.$$

Answer: 73.3.

b) $P(Z > z) = P\left(Z > \frac{x_1 - \mu}{\sigma}\right) = P\left(Z > \frac{57 - 50}{10}\right) = P(Z > 0.7) = 1 - P(Z < 0.7).$

Then from normal table,

$$\mathrm {P} (\mathrm {Z} < 0. 7) = 0. 7 5 8 0.$$

And

$$P (Z > z) = 1 - 0. 7 5 8 = 0. 2 4 2 < 0. 2 5.$$

Hence a score of 57 would earn a stick of gum.

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