PhStat HYPOTHESIS TESTING

The setup

P-VALUE REJECT NULL HYPOTHESIS WHEN 𝑝−𝑣𝑎𝑙𝑢𝑒<𝛼 (𝑙𝑒𝑣𝑒𝑙 𝑜𝑓 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒) DO NOT REJECT NULL HYPOTHESIS WHEN 𝑝−𝑣𝑎𝑙𝑢𝑒≥𝛼 (𝑙𝑒𝑣𝑒𝑙 𝑜𝑓 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒)

Example 1 (𝜇) case1 𝜎 known A manufacturer of sports equipment has developed a new synthetic fishing line that he claims has a mean breaking strength of 8 kilograms with a standard deviation of 0.5 kilogram. Test the hypothesis that  = 8 kilograms against the alternative that   8 kilograms if a random sample of 15 lines is tested and is found to have a mean breaking strength of 7.8 kilograms. Use a 0.01 level significance.

Example 2 (𝜇) case 1 𝜎 unknown but n>30 It is claimed that an automobile is driven on the average less than 20,000 kilometers per year. To test this claim, a random sample of 100 automobile owners is 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 kilometers and standard deviation of 3,900 kilometers? Use 0.01 level of significance.

Example 3 (𝜋) A builder claims that heat pumps are installed in 70% of all homes being constructed today in the city of Richmond. Would you agree with this claim if a random survey of new homes in this city shows that 8 out of 15 had heat pumps installed?

Example 4 ( 𝜎 2 ) A manufacturer of car batteries claims that the life of his batteries have a standard deviation equal to 0.9 year. If a random sample of 10 of these batteries have a standard deviation of 1.2 years, do you think that  > 0.9 year? Use a 0.05 level of significance.

Example 5 (1-2 ) case 2 The following data represent the running times of films produced by two motion picture companies: At the 0.05 level of significance, is there evidence that the average running time of motion pictures produced by company I is longer than those produced by company II? Assume equal variances.   TIME (in minutes) Company I Company II 103 94 110 87 98 97 82 123 92 175 88 118

Example 6 (1-2 ) case 3 Assuming unequal variances in EXAMPLE 5 we have:

Example 7 (1-2 ) case 4 A taxi company is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were equipped radial tires and driven over a prescribed test course. Without changing drivers, the same cars were then equipped with regular belted tires and driven once again over the test course. The gasoline consumption, in kilometers per liter, was recorded as follows: Car 1 2 3 4 5 6 7 8 9 10 11 12 Radial Tires Belted Tires 4.2 4.7 6.6 7.0 6.7 4.5 5.7 6.0 7.4 4.9 6.1 5.2 4.1 4.9 6.2 6.9 6.8 4.4 5.7 5.8 6.9 4.7 6.0 4.9

Example 8 ( 𝜋 1 − 𝜋 2 ) A cigarette manufacturing firm distributes two brands of cigarettes. If it is found that 56 of 200 smokers prefer brand A and that 29 of 150 smokers prefer brand B, can we conclude at the 0.06 level of significance that brand A outsells brand B?

EXAMPLE 9 𝜎 1 2 𝜎 2 2 To verify the assumptions in EXAMPLE 5 we test whether the variances are equal or not:

NOTE 𝑓 1−𝛼 𝑣 1 , 𝑣 2 = 1 𝑓 𝛼 𝑣 2 , 𝑣 1 The f-distribution table provided in class only provides values for 𝑓 𝛼 𝑥,𝑦 so the above formula helps you obtain values for 𝑓 1−𝛼 𝑥,𝑦