The Central Limit Theorem says that no matter what the distribution of the population is, as long as the sample is “large,” meaning of size 30 or more, the sample mean is approximately normally distributed. If the population is normal to begin with then the sample mean also has a normal distribution, regardless of the sample size.
For samples of any size drawn from a normally distributed population, the sample mean is normally distributed, with mean μX-=μ and standard deviation σX-=σ/n, where n is the sample size.
Figure 6.4 Distribution of Sample Means for a Normal Population
A prototype automotive tire has a design life of 38,500 miles with a standard deviation of 2,500 miles. Five such tires are manufactured and tested. On the assumption that the actual population mean is 38,500 miles and the actual population standard deviation is 2,500 miles, find the probability that the sample mean will be less than 36,000 miles. Assume that the distribution of lifetimes of such tires is normal. Solution For simplicity we use units of thousands of miles. Then the sample mean X- has mean μX-=μ=38.5 and standard deviation σX-=σ/n=2.5/5=1.11803. Since the population is normally distributed, so is X-, hence P(X-<36)=P(Z<36−μX-σX-)=P(Z<36−38.51.11803)=P(Z<−2.24)=0.0125That is, if the tires perform as designed, there is only about a 1.25% chance that the average of a sample of this size would be so low.
An automobile battery manufacturer claims that its midgrade battery has a mean life of 50 months with a standard deviation of 6 months. Suppose the distribution of battery lives of this particular brand is approximately normal.
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