Question

1.   How large of a sample is required if we want to estimate the mean price of a five year old Corvette within +/- $500 with 90% confidence if the standard deviation is known to be $3100.

2.   Several factors are involved when creating confidence intervals, such as sample size, the level of confidence and margin of error. Which of the following statements are true.

a.   For a given sample size, reducing the margin of error will mean lower confidence.
b.   For a specified confidence interval. Larger samples provide smaller margins of error.
c.   For a given confidence level, halving the margin of error requires a sample twice as large.

3.   When conducting a test for the difference of means for two independent populations x1 and x2, what alternate hypothesis would indicate that the mean of x2 population is smaller than that of the x1 population?

a)   H1: µ1 ‹ µ2
b)   H1: µ1 ≠ µ2
c)   H1: µ1 › µ2
d)   H1: µ1 = µ2

4.   A researcher wants to estimate the average amount of money a person spends on buying lottery tickets each week. A sample of 50 people who buy lottery tickets was found with a mean of $15 and a standard deviation of 3.4. Find a 90% confidence interval of the population mean. Explain in laymen terms what this interval represents.

5.   Discuss the basic components of a hypothesis test.

6.   Explain how to conclude when you would reject the null hypothesis based on a p-value. Give a numeric example.

7.   A medical rehabilitation foundation reports that average cost of rehabilitation for stroke victims is $23,672. A researcher wishes to find out the average costs of rehabilitation by selecting a random sample of 35 stroke victims and finds their average cost to be $24,226. The standard deviation of the population is $3251. At an alpha level of 0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from $23,672?

8.   A medical researcher wishes to see whether the pulse rate of smokers are higher than the pulse rates of non-smokers. Samples of 100 smokers and 100 nonsmokers are selected. The results are shown below. Can the researcher conclude at an alpha = 0.05, that smokers have higher pulse rates than nonsmokers?

SmokersNon.                                 Smokers
Mean = 90.                                     Mean = 88
Standard Deviation = 5.         Standard Deviation = 6

Answer 1

1) Margin of errror = E = 500

Level of significance = 1 - 0.90 = 0.10

So, the critical value of z for two tailed test at 0.10 level of significance = = 1.65

(It can be obtained from the z table by finding the z corresponding to the area close to 0.10/2)

Standard deviation = = 3100

Hence, the sample size required will be -

= 104.6

~ 105

2) a is false as reducing the margin of errror increases the confidence.

b is true as we increases the sample size, the margin of errror increases, since,

c is also False because, since , so, if E is halved, sample size has to be 4times large than before.

3) Mean of X2 is smaller than X1, for this alternative will be (population mean of second population is less than population mean of the first population), or,

So, c is the correct option.

4) 90% confidence interval for the population mean is given by -

where, is the sample mean = 15

n is the sample size = 50

is the standard deviation = 3.4

is the critical value of z for two tailed test at 0.10 level of significance = 1.65

So, the confidence interval will be -

= [ 14.21, 15.78]

So, we are 90% confident that the population mean will lie between 14.21 and 15.78

5) Basic components of the hypothesis test are -

Framing null and alternative hypothesis

Checking validation of assumption of sampling distribution and deciding test to be used accordingly.

Defining and finding the test statistic.

Finding p value or critical value for comparison.

Writing the Conclusion.

6) If P value < level of significance, we Reject the null hypothesis

Otherwise we do not Reject it.

Suppose we wish to compare effect of two drugs.

Null hypothesis : There is no significant difference in effect of two drugs.

Alternative hypothesis : There is significant difference between the effect of the drugs.

And P value comes out to be 0.1608 (say)

and our level of significance = 0.05

Now, since, P value > level of significance, we may not reject the null hypothesis, so, we will conclude that there is no significant difference between the effect of two drugs.