MAST20005/MAST90058: Week 7 Problems
Some useful information for many of the problems is shown at end of this problem sheet.
1. A ball is drawn from one of two bowls. Bowl A contains 100 red balls and 200 white balls; Bowl B contains 200 red balls and 100 white balls. Let p denote the probability of drawing a red ball from the chosen bowl. Then p is unknown as we don’t know which bowl is being used. We shall test the simple null hypothesis, H0 : p = 1/3, against the simple alternative, H1 : p = 2/3. We draw three balls at random with replacement from the selected bowl. Let X be the number of red balls drawn. Let the critical region be x ∈ {2, 3}. What are the probabilities α and β respectively of Type I and Type II errors?
2. LetY ∼Bi(100,p). TotestH0:p=0.08againstH1:p<0.08,werejectH0 ifY 6. (a) Determine the significance level α of the test.
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(b) Find the probability of a Type II error if, in fact, p = 0.04.
3. If a newborn baby has a birth weight that is less than 2500 grams we say the baby has a low birth weight. The proportion of babies with low birth weight is an indicator of nutrition for the mothers. In the USA, approximately 7% of babies have a low birth weight. Let p be the proportion of babies born in the Sudan with low birth weight. Test the null hypothesis H0 : p = 0.07 against the alternative H1 : p > 0.07. If y = 23 babies out of a random sample of n = 209 babies had low birth weight, what is your conclusion at the following significance levels:
(a) α = 0.05?
(b) α = 0.01?
(c) Find the p-value of this test. (Recall the p-value is the probability of the observed value or something more extreme when the null hypothesis is true).
4. Let pm and pf be the respective proportions of male and female white crowned sparrows that return to their hatching site. Give the endpoints for a 95% confidence interval for pm − pf , given that 124 out of 894 males and 70 out of 700 females returned. Does this agree with the conclusion of the test of H0 : pm = pf against H1 : pm ̸= pf with α = 0.05?
5. Vitamin B6 is one of the vitamins in a multivitamin pill manufactured by a pharmaceutical company. The pills are produced with a mean of 50 milligrams of vitamin B6 per pill. The company believes there is a deterioration of 1 milligram per month, so that after 3 months they expect that μ = 47. A consumer group suspects that μ < 47 after 3 months.
(a) Define a critical region to test H0 : μ = 47 against H1 : μ < 47 with a significance level of α = 0.05 based on a random sample of size n = 20 and assuming a normal distribution.
(b) If the 20 pills resulted in a mean of x ̄ = 46.94 and a standard deviation of s = 0.15, what is your conclusion?
(c) Give bounds for the p-value of this test.
6. Let X represent the height of professional female volleyball players. Assume that X ∼ N(μ,σ2) approximately. Suppose it is known that μ = 1.9 metres worldwide. Do Aus- tralian female players differ from this? We explore this using a random sample of size n = 9.
(a) Define the null hypothesis.
(b) Define the alternative hypothesis.
(c) Define a critical region for which α = 0.05.
(d) Calculate the value of the test statistic if x ̄ = 2.05 and s = 0.17.
(e) What is your conclusion?
(f) Give bounds for the p-value of this test.
7. In May, the weights of 2-kilogram boxes of laundry detergent had a mean of 2.13 kilograms with a standard deviation of 0.095. The goal was to decrease the standard deviation. The company decided to adjust the filling machines and then test H0 : σ = 0.095 against H1:σ<0.095. InJune,arandomsampleofsizen=20gavex ̄=2.10ands=0.065.
(a) At an α = 0.05 significance level, was the company successful? (b) What is an approximate p-value for this test?
8. The World Health Organisation collects air quality data around the world, which includes a measurement of suspended particles in μ g/m3. Let X and Y equal the concentration of suspended particles in the city centres of Melbourne and Sydney, respectively. Using n = 13 observations of X and m = 16 observations of Y, we shall test H0: μX = μY against H1 : μX ̸= μY using a significance level of 5%.
(a) Define the test statistic and the critical region assuming the variances are equal.
(b) If x ̄ = 72.9, sX = 25.6, y ̄ = 81.7 and sY = 28.3, calculate the value of the test statistic and state your conclusion.
(c) Give bounds for the p-value of this test.
(d) Test whether the assumption of equal variances is valid. Let α = 0.05.
Some potentially helpful R output:
> dbinom(0:3, 3, 1/3)
[1] 0.29629630 0.44444444 0.22222222 0.03703704
> dbinom(0:3, 3, 2/3)
[1] 0.03703704 0.22222222 0.44444444 0.29629630
> pnorm(c(-0.737, -0.553, 1.276, 1.531, 2.269))
[1] 0.2305612 0.2901317 0.8990222 0.9371153 0.9883658
> p1 <- c(0.75, 0.9, 0.95, 0.975, 0.99)
> qnorm(p1)
[1] 0.6744898 1.2815516 1.6448536 1.9599640 2.3263479
> qt(p1, 27)
[1] 0.683685 1.313703 1.703288 2.051831 2.472660
> qt(p1, 20)
[1] 0.6869545 1.3253407 1.7247182 2.0859634 2.5279770
> qt(p1, 19)
[1] 0.6876215 1.3277282 1.7291328 2.0930241 2.5394832
> qt(p1, 8)
[1] 0.7063866 1.3968153 1.8595480 2.3060041 2.8964594
> p2 <- c(0.025, 0.05, 0.95, 0.975)
> qchisq(p2, 19)
[1] 8.906516 10.117013 30.143527 32.852327
> qf(p2, 12, 15)
[1] 0.3147424 0.3821387 2.4753130 2.9632824
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