Statistics, EXT1 EQ-Bank 26

A research team is investigating the amount of sleep obtained by Year 12 students. Previous studies indicate that the sleep time of Year 12 students has a population mean of 7.4 hours and a population standard deviation of 2.42 hours.

A random sample of 100 Year 12 students is selected.

Using the normal distribution table (included), determine the probability that the mean sleep time of the sample is less than 7 hours. Give your answer correct to four decimal places. (3 marks)

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\(4.94 \%\)

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

\(\text{The sample mean,}\ \overline{X}, \text{for random samples of size 100 is}\)

\(\text{approximately normally distributed, where:}\)

\(\mu=7.4\ \ \text{and}\ \ \sigma=\dfrac{2.42}{\sqrt{n}}=\dfrac{2.42}{\sqrt{100}}\)

\(\text{By the central limit theorem:}\)

\(Z=\dfrac{\overline{X}-7.4}{\frac{2.42}{\sqrt{100}}} \sim N(0,1)\)

\(Z=\dfrac{7-7.4}{\frac{2.42}{\sqrt{100}}}=-1.65 \ \text{(2 d.p.)}\)

\(\text{Using Normal Distribution Table of Values:}\)

\(P(\overline{X} < 7)=P(Z < -1.65)=0.0494=4.94 \%\)