Statistics, EXT1 EQ-Bank 25

In a large school, the average amount of money spent per student per day at the canteen is $8 with a standard deviation of 6.5 .

At the end of each day, 50 randomly chosen students are asked how much they spent at the canteen on that day.

Use the standard normal distribution table (included) to find the probability that the sample mean on a particular day is greater than $10. (3 marks)

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$1.46 \%$

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$\text{Sample size is > 30} \ \ \Rightarrow \ \ \text{Central Limit Theorem applies}$

$\text{The sample mean,}\ \overline{X}, \text{for random samples of size 50 is}$

$\text{approximately normally distributed, where:}$

$\mu=8\ \ \text{and}\ \ \sigma=\dfrac{6.5}{\sqrt{n}}=\dfrac{6.5}{\sqrt{50}}$

$\text{By the central limit theorem:}$

$Z=\dfrac{\overline{X}-8}{\frac{6.5}{\sqrt{50}}} \sim N(0,1)$

$Z=\dfrac{10-8}{\frac{6.5}{\sqrt{50}}}=2.18 \ \text{(2 d.p.)}$

$\text{Using Normal Distribution Table of Values:}$