Question

Z-test and Confidence Intervals

1a) Allister Ride Company  estimate that the average daily loss from those illegally riding without tickets is at least (>=) $200, but wants to determine the accuracy of this statistic. The company researcher takes a random sample of losses over 64 days and finds that = $198 and s = $15.

a) Test at α = 0.01.

b) Construct a 99% confidence interval

b) Kids Home Care claims that they use more than 110 diapers daily. The manager takes a random sample of used diapers use 61 days (2 months) and finds that the sample mean is 108 diapers with a standard deviation of 17.

a) Test at α = 0.06

b) Construct a 90% confidence interval

2) A college wants to know the lower limit of a two-sided 95% confidence interval for SAT scores on the Math section (range equals 200 to 800).

They sampled 400 students and found that the sample mean was 560 and the sample standard deviation was 35.

Answer 1

here sample standard deviation is given instead of population standard deviation, so we should use t-value (instead of z-value) for answering the question. if you want using z-value please respond back at earliest for correction.

(1a) here we want to test the null hypothesis H0: <=200 and alternate hypothesis Ha:>=200 ( this left-one tailed test)

statistic t=(-)/(s/sqrt(n))=(198-200)/(15/sqrt(64))=-1.07 is more than critical t(0.01,63)=-2.39 so we fail to reject H0 and conclude that loss is not atleast 200.

n=	64
sample mean=	198.000
s=	15.000

the two sided

(1-alpha)*100% confidence interval for population mean=sample mean±t(alpha/2,n-1)*s/sqrt(n)

99% confidence interval for population mean=mean±t(0.05/2, n-1)*s/sqrt(n)=198±2.656*15/sqrt(64)=198±4.980=(193.02,202.98)

two-sided	t-value	margin of error	lower limit	upper limit
99% confidence interval	2.656	4.980	193.020	202.980

(second part)

here we want to test the null hypothesis H0: <=110 and alternate hypothesis Ha:>110

statistic t=(-)/(s/sqrt(n))=(108-110)/(17/sqrt(61))=-0.92 is more than critical t(0.01,63)=-1.58, so we fail to reject H0 and conclude Kids Home Care claims that they use more than 110 diapers daily is not true.

two-sided	t-value	margin of error	lower limit	upper limit
94% confidence interval	1.917	4.173	103.827	112.173

(third part) lower limit=556.56

n=	400
sample mean=	560.000
s=	35.000

	t-value	margin of error	lower limit	upper limit
95% confidence interval	1.966	3.440	556.560	563.440