. 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.
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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 Ha: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 onetailed and twotailed 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. Pvalue
The pvalue 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 Ha.
Ho: Total space is 250 square feet.
Ha: 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: pA ≥ pB and Ha: pA < pB 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
effective as 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. CoughaLot 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):
5.9
5 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.
H 0 : µ=6∧H a : µ≠ 6
b. Calculate the test statistic.
Sample mean = 5.989 so, Test Statistic = 5.989−6
=¿
0.3
0.1160
√10 c. Find the Pvalue.
Pvalue = P(Z > 0.1160) = 0.9077
d. What is the conclusion?
Pvalue 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 = (2017)/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 = (1214)/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 Orama 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.
H 0 : µ ≥ 400∧H a : µ>400 Z score = 385−400
35
√100 = 4.2857 PValue = P(Z > 4.2857) = 1.00
Large pvalue 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.
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