A type i error is the event of rejecting the null

Question

Answer the following problems showing your work and explaining (or analyzing) your results.

1.   Explain Type I and Type II errors. Use an example if needed.

A Type I error is the event of rejecting the null hypothesis when it is true. And a Type II error is not rejecting a false hypothesis.

For example suppose we want to test,

H₀: µ = 150 and H_a:µ ≠150

And we rejected the null hypothesis based on the sample data though the true population mean is not significantly different from 150 then we would committed a Type I error.

And in this case if we don't reject the null hypothesis based on the data though the true population mean is significantly different from 150 we would commit a Type II error.

2.   Explain a one-tailed and two-tailed test. Use an example if needed.

A one tail test is the hypothesis test where the alternative hypothesis is one sided i.e. alternative hypothesis contains "<" or ">".

A two tailed test is where the alternative hypothesis is two sided i.e. alternative hypothesis contains "≠".

3.   Define the following terms in your own words.

• Null hypothesis

Null hypothesis is the unbiased hypothesis (i.e. always contains "=" or "≤" or "≥" sign) which can't be proved true. We take an alternative and try to reject the null hypothesis to show that the alternative hypothesis is true.

• P-value

The p-value is the probability of getting the test statistic as obtained or extreme when the null hypothesis is true.

• Critical value

The critical value is a value of the testing distribution at some specific significance level which I used to test whether the null hypothesis is rejected or not.

• Statistically significant

If we reject the null hypothesis we conclude that the result is statistically significant.

4.   A homeowner is getting carpet installed. The installer is charging her for 250 square feet. She thinks this is more than the actual space being carpeted. She asks a second installer to measure the space to confirm her doubt. Write the null hypothesis H_o and the alternative hypothesis H_a.

H_o: Total space is 250 square feet.

H_a: Total space is less than 250 square feet.

5.   Drug A is the usual treatment for depression in graduate students. Pfizer has a new drug, Drug B, that it thinks may be more effective. You have been hired to design the test program. As part of your project briefing, you decide to explain the logic of statistical testing to the people who are going to be working for you. 

• Write the research hypothesis and the null hypothesis.

H₀: p_A≥p_B and H_a: p_AB

• Then construct a table like the one below, displaying the outcomes that would constitute Type I and Type II error.

	True State
Our Decision		H0 is true: Drug B is not more effective than Drug A	H1 is true: Drug B is  more effective than Drug A
	H0 is true: Drug B is as effectiveas Drug A	No error	Type II error: Decision is the Drug B is not more effective but actually it is
	H1 is true: Drug B is more effective than Drug A	Type I error: Decision is the Drug B is more effective but actually it is not	No error
	Consequences of Error:	Many patients will suffer due to use of less effective drug	A better drug would be unused and manufacturers would lose money.

• Write a paragraph explaining which error would be more severe, and why.

Here Type I error is more severe as in that case many patients will use less effective drug and that may cause even death. On the other hand Type II error means the manufacturer will only lose some money.

6.   Cough-a-Lot children's cough syrup is supposed to contain 6 ounces of medicine per bottle. However since the filling machine is not always precise, there can be variation from bottle to bottle. The amounts in the bottles are normally distributed with σ = 0.3 ounces.  A quality assurance inspector measures 10 bottles and finds the following (in ounces):

5.95

6.10

5.98

6.01

6.25

5.85

5.91

6.05

5.88

5.91

Are the results enough evidence to conclude that the bottles are not filled adequately at the labeled amount of 6 ounces per bottle?

a.   State the hypothesis you will test.

b.   Calculate the test statistic.

Sample mean = 5.989 so,

Test Statistic = -0.1160

c.    Find the P-value.

P-value = P(|Z| > 0.1160) = 0.9077

d.   What is the conclusion?

P-value is large so do not reject the null hypothesis concluding that the bottles are filled adequately.

7.   Calculate a Z score when X = 20, μ = 17, and σ = 3.4.

Z score = (20-17)/3.4 = 0.8824

8.   Using a standard normal probabilities table, interpret the results for the Z score in Problem 7.

The Z score is Problem 7 implies that the value 20 is 0.8824 standard deviation above the mean of 17.

9.   Your babysitter claims that she is underpaid given the current market. Her hourly wage is $12 per hour. You do some research and discover that the average wage in your area is $14 per hour with a standard deviation of 1.9. Calculate the Z score and use the table to find the standard normal probability. Based on your findings, should you give her a raise? Explain your reasoning as to why or why not.

Z score = (12-14)/1.9 = -1.05

P(Z< -1.05) = 0.1469

Only 14.69% of the other baby sitters are earning $12 or less so I should give her a raise.

10.Tutor O-rama claims that their services will raise student SAT math scores at least 50 points. The average score on the math portion of the SAT is μ = 350 and σ = 35. The 100 students who completed the tutoring program had an average score of 385 points. Is the average score of 385 points significant at the 5% level? Is it significant at the 1% level? Explain why or why not.

Z score =  = -4.2857

P-Value = P(Z > -4.2857) = 1.00

Large p-value implies that we should reject not reject the null hypothesis at both 5% and 1% significance level concluding that the average score of 385 points is not significant at the 5% level as well as 1% level.