Question

8. Spring exam 2013 The desired amount of SiO2 in 100 grams of a certain type of aluminous cement is 5.5 grams. A production facility wants to test whether the amount of SiO2 in its production is as desired. A total of 16 independent 100 grams samples were collected and they analyzed The sample mean of these samples was 5,25 grams. Assume that the amount of SiO2 in 100 grams follows a normal distribution with mean ,1 and a known standard deviation ơ-0.3. Test the null hypothesis μ = 5.5 versus the alternative hypothesis μメ5.5. Use a significance level a 0.05 a) What is the value of the test statistic? b) Assuming that 5.6 and that the number of samples is n 16, what is the probability of rejecting the null hypothesis? c Assuming that 5.6, how many samples are needed to ensure that the probability of type II error is equal to 0.10 or just slighty less than 0.10?

Answer 1

a)

null hypothesis:Ho Alternate Hypothesis:Ha 5.5 5.5 for 0.05 level with two tailed test, critical value of z- Decision rule 1.960 reject Ho if absolute value of test statistic zl 1.96 population mean sample mean sample size population std deviation 5.5 5.25 16.00 0.300 μέ standard error of means ơ,-ơ//n- 0.0750 test statistic z (X-μ)/σ,- -3.33 =

b)

c)

Hypothesized mean true mean standard deviation for 0.05 level and two tailed test critival value Ζα/2 for 0.1 level of type Il error critival value ZB required sample sizen 5.5 5.6 0.3 1.9600 1.2816 95.00