Given the normal distribution with μ = 49 and σ = 8, find the probability that X assumes a value:
Less than 45
More than 50
Example 2 The achievement sores for a college entrance examination are normally distributed with the mean 75 and standard deviation equal to 10. What fraction of the scores would one expect to lie between 70 and 90.
As the sample size increases , the distribution of the sample average becomes less and less variable.
Hence the sample average X bar approaches the value of the population mean μ .
Example 3 An electrical firm manufactures light bulbs that have a length of life normally distributed with mean and standard deviation equal to 500 and 50 hours respectively. Find the probability that a random sample of 15 bulbs will have an average life ofless than 475 hours.
Testing a Hypothesis on the Population Mean Z = X - μ 0 σ /√n t = X - μ 0 S /√n υ = n - 1 H 0 Test Statistic H a Critical Region σ known μ = μ 0 μ < μ 0 μ > μ 0 μ ≠ μ 0 z < -z α z > z α |z| > z α /2 σ unknown μ = μ 0 μ < μ 0 μ > μ 0 μ ≠ μ 0 t < -t α t > t α |t| > t α /2
critical value test statistic Do not reject H 0
Example 4 It is claimed that an automobile is driven on the average of less than 25,000 km per year. To test this claim, a random sample of 100 automobile owners are asked to keep a record of the kilometers they travel. Would you agree with this claim if the random sample showed an average of 23,500 km and a standard deviation of 3,900 km? Use 0.01 level of significance.