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The gestation period of cats has a mean of 66 days and a variance of 9 days\(^2\).
A sample of 32 cats is chosen at random.
Using the Normal Distribution Table of Values, determine the probability that the sample has an average gestation period greater than 65 days. (3 marks)
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\(0.9706\)
\(\text{Sample size is > 30} \ \ \Rightarrow \ \ \text{CLT applies}\)
\(\overline{X} \sim N\left( \mu, \dfrac{\sigma^2}{n}\right) \sim N\left( 66, \dfrac{9}{32}\right)\)
\(\sigma_{\bar{X}} = \dfrac{3}{\sqrt{32}} \)
\(\text{By the central limit theorem:}\)
\(Z=\dfrac{\overline{X}-66}{\frac{3}{\sqrt{32}}}\sim N(0,1)\)
\(Z=\dfrac{65-66}{\frac{3}{\sqrt{32}}}=-1.89 \ \text{(2 d.p.)}\)
\(\text{Using Normal Distribution Table of Values:}\)
| \(P(\overline{X}>65)\) | \(=P(Z>-1.89)\) |
| \(=1-P(Z\leq -1.89)\) | |
| \(=1-0.0294\) | |
| \(=0.9706\) |