In
Calculation formula for the test statistic (z-score or t-score)
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What p-value(s) describe strong evidence against the null hypothesis?
.01 < P ≤ .05
What p-value(s) describe moderate evidence against the null hypothesis?
.05 < P ≤ .10
Determine the p-value. Z = 2.03 right-tailed test
0.0212
Determine the p-value. Z = -1.66 two-tailed test
0.0970
Determine the p-value. Z = -0.74 left-tailed test
0.2296
Determine the p-value. Z = 0.52 two-tailed test
0.6030
Determine the p-value. Z = -0.31 right-tailed test
0.6217
Determine the p-value. Z = 1.16 left-tailed test
0.8770
According to Scarborough Research, more than 85% of working adults commute by car. Of all U.S. cities, Washington, D.C. and New York City have the longest commute times. A sample of 30 commuters in the Washington, D.C. area yielded the following commute times, in minutes (data set, x-bar=27.97 minutes, s=10.04 minutes). (1) Find a 90% confidence interval for the mean commute time of all commuters in Washington, D.C. (2) Interpret your answer from part (1).
(1) (24.855, 31.085) (2) We are 90% confident that the average commute time of all commuters in Washington, D.C. is between 24.855 minutes and 31.085 minutes.
Would it be appropriate to use a t-interval for a sample size of 15? Explain.
It would not be appropriate because the assumption of a normal population or a large sample is not met. We know nothing of the population and the sample is small.
On the calculator (TI-84), how do you find the area to the left of a particular z-score?
NORMALCDF (-1000, Z-score, 0, 1)
On the calculator (TI-84), how do you find the area to the right of a particular z-score?
NORMALCDF (Z-score, 1000, 0, 1)
Type II Error
Not rejecting the null hypothesis when it is in fact false.
What p-value(s) describe very strong evidence against the null hypothesis?
P ≤ .01
What p-value(s) describe weak or no evidence against the null hypothesis?
P-value > .10
Type I Error
Rejecting the null hypothesis when it is in fact true
Given: n=45 x-bar = 14.68 ϑ = 4.2 ∝ = .01 H₀: µ = 18 H₁: µ < 18 Reject or do not reject null hypothesis?
Test statistic: z = -5.3 Critical value: -2.326 Conclusion: -5.3 is in the rejection region Interpretation: There is sufficient evidence to conclude that ... the alternative hypothesis is true.
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose that the specifications state that the mean strength of welds should exceed 100 lb/ in². The inspection team decides to test H₀: µ = 100 versus H₁: µ > 100. Explain why this alternative hypothesis was chosen rather than µ < 100.
The alternative hypothesis was chosen because the mean strength of welds should be greater than 100, thus the alternative hypothesis H₁: µ > 100. The primary concern of the research is to decide whether the population mean is greater than the specified value (and meets specifications).
alternative hypothesis
The alternative to the null hypothesis
Explain the difference in the formulas for the standardized and the studentized version of x-bar.
The denominator of the standardized version of x-bar (z-score) uses the population standard deviation, ϑ, whereas the denominator of the studentized version of x-bar (t-score) uses the sample standard deviation, s.
Two t-curves have degrees of freedom, 12 and 20, respectively. Which one more closely resembles the standard normal curve? Explain your answer.
The df=20 because as the number of degrees of freedom increases, t-curves look increasingly like the standard normal curve.
null hypothesis
The hypothesis to be tested
If you have a p-value of 0.0168 and a z-score of +/- 2.39, interpret the meaning of these values in context.
The probability of getting a z-score more extreme than +/- 2.39 is 0.0168.
Define significance level
The probability of making a Type I error, that is, of rejecting a true null hypothesis.
What is the p-value of a hypothesis test?
The probability of observing a value of the test statistic as extreme or more extreme than that observed. By extreme we mean "far from what we would expect to observe if the null hypothesis is true."
hypothesis test
The problem in a hypothesis test is to decide whether the null hypothesis should be rejected in favor of the alternative hypothesis.
non-rejection region
The set of values for the test statistic that lead us not to reject H₀
rejection region
The set of values for the test statistic that lead us to reject H₀ (tail or tails of the distribution)
Define non-rejection region
The set of values for the test statistic that leads to non-rejection of the null hypothesis.
Define rejection region
The set of values for the test statistic that leads to rejection of the null hypothesis.
Define test statistic
The statistic used as a basis for deciding whether the null hypothesis should be rejected
Define critical values
The values of the test statistic that separate the rejection and non-rejection regions. A critical value is considered part of the rejection region.
What does the z or t test statistic tell us?
The z or t test statistic tells us how far x-bar is from µ in standard deviations (i.e. the number of standard deviations from the mean). It is the statistic used as a basis for deciding whether the null hypothesis should be rejected.
Describe the meaning of P-value of a hypothesis test
To obtain the P-value of a hypothesis test, we assume that the null hypothesis is true and compute the probability of observing a value of the test statistic as extreme as or more extreme than that observed. By extreme we mean "far from what we would expect to observe if the null hypothesis is true." We use the letter P to denote the P-value. The p-value is the area beyond the test statistic in either direction.
True or False. The length of a confidence interval can be determined if you know only the margin of error.
True
True or False. The margin of error can be determined if you know only the length of the confidence interval.
True
True or False: The confidence interval can be obtained if you know only the margin of error and the sample mean.
True
True or False: The p-value is the smallest significance level for which the observed sample data result in rejection of the null hypothesis.
True
True or False: For a fixed sample size, decreasing the significance level of a hypothesis test results in an increase in the probability of making a Type II error.
True. For a fixed sample size, the smaller you specify the significance level ∝, the larger will be the probability β, of not rejecting a false null hypothesis.
True or False: If it is important not to reject a true null hypothesis, the hypothesis test should be performed at a small significance level.
True. The significance level ∝ of a hypothesis test is the probability of making a Type I error (rejecting a true null hypothesis). If this is important, the lower the probability of making such an error the better; thus you should use a small significance level.
When does the p-value provide evidence against the null hypothesis?
When the p-value is less than or equal to the significance level, ∝
critical values
the boundaries for the rejection/non-rejection regions
Type I Error probability
the probability of a Type I error, denoted ∝, also called the significance level of the hypothesis test
Type II Error probability
the probability of a Type II error, denoted β - a Type II error occurs if the test statistic falls in the non-rejection region when in fact the null hypothesis is false.
significance level
the probability of making a Type I error, that is, of rejecting a true null hypothesis (denoted ∝)
test statistic
the z-score (or t-score) that determines if an average is unusual or not
An automobile manufacturer who wishes to advertise that one of its models achieves 30 mpg decides to carry out a fuel efficiency test. Six non-professional drivers are selected and each one drives a car from Phoenix to Los Angeles. The resulting fuel efficiencies in mpg are: 27.2, 29.3, 31.2, 28.4, 30.3, 29.6. Assuming that the fuel efficiency is normally distributed, do the data contradict the claim that the true average fuel efficiency is at least 30 mpg? Assume ∝ = 0.05.
(1) H₀: µ ≥ 30 mpg H₁: µ < 30 mpg (2) ∝ = 0.05 (3) test statistic t = -1.16 (4) p-value = .1493 (t-test on calculator) (5) compare p-value to ∝ .1493 ≤ 0.05 ? NO Do not reject H₀ (6) There is not enough evidence to conclude that the average fuel efficiency in mpg is less than 30 mpg.
Steps for Hypothesis Tests for One Population Mean when ϑ is known (6)
(1) State the null and alternative hypotheses (2) Decide on a value for ∝ (significance level) (3) Compute the test statistic Z (4) Find the critical values (5) Conclusion (6) Interpretation
The method for computing the sample size required to obtain a confidence interval with a specified confidence level and margin of error - the number resulting from the formula should be rounded up to the nearest whole number. (1) Why do you want a whole number? (2) Why do you round up instead of down?
(1) The sample size cannot be a fraction. (2) The result (n) is the smallest value that will provide the required margin of error. If the number were rounded down, the sample size would not be sufficient to ensure the required margin of error.
Identify the two types of incorrect decisions in a hypothesis test. For each incorrect decision, what symbol is used to represent the probability of making that type of error?
(1) Type I - rejecting a true null hypothesis (symbol ∝) (2) Type II - not rejecting a false null hypothesis (symbol β)
2 methods for determining whether to reject or not reject the null hypothesis
(1) compare the test statistic to the critical values; where the test statistic falls (rejection region or non-rejection region) (2) compare the p-value to ∝ If the p-value is low, H₀ must go! If the p-value ≤ ∝, reject H₀ If the p-value > ∝, do not reject H₀
State two reasons why including the p-value is prudent when you are reporting the results of a hypothesis test.
(1) it allows you to assess significance at any desired level (2) it permits you to evaluate the strength of the evidence against the null hypothesis
Determine the strength of the evidence against the null hypothesis: (a) p = 0.06 (b) p = 0.35 (c) p = 0.027 (d) p = 0.004
(1) moderate (2) weak or none (3) strong (4) very strong
Given: safety limit set for cadmium in dry vegetables at 0.5 ppm. A hypothesis test is to be performed to decide whether the mean cadmium level in Bp mushrooms is greater than the government's recommended limit. (1) Determine the null hypotheses, (2) Determine the alternative hypothesis, (3) classify the hypothesis test as two tailed, left tailed, or right tailed.
(1) null hypothesis H₀: µ = .5 ppm (2) alternative hypothesis H₁: µ > .5 ppm (3) right tailed test
The recommended dietary allowance (RDA) of iron for adult females under the age of 51 is 18 milligrams (mg) per day. A hypothesis test is to be performed to decide whether adult females under the age of 51 are, on average, getting less than the RDA of 18 mg of iron. (1) Determine the null hypothesis, (2) Determine the alternative hypothesis, (3) classify the hypothesis test as two tailed, left tailed, or right tailed.
(1) null hypothesis H₀: µ = 18 mg (2) alternative hypothesis H₁: µ < 18 mg (3) left tailed test
A confidence interval for a population mean has a margin of error of 3.4. If the sample mean is 52.8, obtain the confidence interval.
(49.4, 56.2)
What mathematical signs are allowed in the alternative hypothesis?
≠, <, >
Infants treated for pulmonary hypertension, called the PH group, were compared with those not so treated, called the control group. One of the characteristics measured was head circumference. The mean head circumference of the 10 infants in the PH group was 34.2 cm. (1) Assuming that head circumferences for infants treated for PH are normally distributed with standard deviation 2.1 cm, determine a 90% confidence interval for the mean head circumference of all such infants. (2) Obtain the margin of error, E, for the confidence interval you found in part (1). (3) Explain the meaning of E in this context in terms of the accuracy of the estimate. (4) Determine the sample size required to have a margin of error of 0.5 cm with a 95% confidence level.
(1) (33.108, 35.292) (2) E = 1.1 cm (3) We can be 90% confident that the error made in estimating µ by x-bar is at most 1.1 cm. (4) 68
The following data are airborne times for United Airlines flight 448 from Albuquerque to Denver on 10 randomly selected days: 57, 54, 55, 51, 56, 48, 52, 51, 59, 59 . (1) Compute and interpret a 90% confidence interval for the mean airborne time for flight 448. (2) Based on your interval in part (1), if flight 448 is scheduled to depart at 10 a.m., what would you recommend for the published arrival time? Explain.
(1) (52.07, 56.33) (2) Recommend an arrival time of 10:57 a.m., so that 0% of the flights would be late.
A data set gives the additional sleep in hours obtained by a sample of 10 patients using laevohysocyamine hydrobromide (with xbar=2.33 hr, s=2.002 hr). (1) Obtain and interpret a 95% confidence interval for the additional sleep that would be obtained on average for all people using laevohysocyamine hydrobromide. (2) Was the drug effective in increasing sleep? Explain your answer.
(1) 0.90 hr to 3.76 hr We can be 95% confident that the additional sleep that would be obtained on average for all people using the drug is somewhere between 0.90 hr and 3.76 hr. (2) It appears so, because, based on the confidence interval, we can be 95% confident that the mean additional sleep is somewhere between 0.90 hr and 3.76 hr and that, in particular, the mean is positive.
A variable has a mean of 100 and a standard deviation of 16. 4 observations of this variable have a mean of 108 and a sample standard deviation of 12. Determine the observed value of the: (1) standardized version of x-bar (2) studentized value of x-bar
(1) 1 (2) 1.33
The p-value for a hypothesis test is 0.083. For each of the following significance levels, decide whether the null hypothesis should be rejected: (1) ∝ = 0.05 (2) ∝ = 0.10 (3) ∝ = 0.06
(1) Do not reject (0.083 > 0.05) (2) Reject (0.083 ≤ 0.10) (3) Do not reject (0.083 > 0.06)
Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The Czech and Slovak governments have set a safety limit for cadmium in dry vegetables at 0.5 parts per million (ppm). A random sample of the edible mushroom Boletus pinicola with the resulting data: 0.24, 0.59, 0.62, 0.16, 0.77, 1.33, 0.92, 0.19, 0.33, 0.25, 0.59, 0.32 . At the 5% significance level, do the data provide sufficient evidence to conclude that the mean cadmium level in Boletus pinicola mushrooms is greater than the government's recommended limit of 0.5 ppm? Assume that the population standard deviation of cadmium levels in Boletus pinicola mushrooms is 0.37 ppm. (Note: The sum of the data is 6.31 ppm). USE THE P-VALUE APPROACH.
(1) H₀: µ = 0.5 ppm H₁: µ > 0.5 ppm (2) ∝ = 0.05 (3) test statistic: Z = 0.24 (4) p-value =0.4044 (calculator t-test) (5) conclusion: compare p-value to ∝: Is 0.4044 ≤ 0.05? No. Do not reject. (6) There is not sufficient evidence to conclude that the average cadmium level in the mushrooms is greater than .5 ppm.
According to the Bureau of Crime Statistics and Research of Australia, the mean length of imprisonment for motor-vehicle theft offenders in Australia is 16.7 months. You want to perform a hypothesis test to decide whether the mean length of imprisonment for motor-vehicle theft offenders in Sydney differs from the national mean in Australia. (1) Determine the null hypothesis, (2) Determine the alternative hypothesis, (3) classify the hypothesis test as two tailed, left tailed, or right tailed.
(1) H₀: µ = 16.7 months (2) H₁: µ ≠ 16.7 months (3) two tailed test
According to Communications Industry Forecast & Report, the average person watched 4.66 hours of television per day in 2002. A random sample of 20 people gave the number of hours of television watched per day for last year (x-bar = 4.835 hours, s = 2.291 hours). At the 10% significance level, do the data provide sufficient evidence to conclude that the amount of television watched per day last year by the average person differed from that in 2002? USE THE P-VALUE APPROACH.
(1) H₀: µ = 4.66 hours H₁: µ ≠ 4.66 hours (2) ∝ = 0.10 (3) Test statistic t = 3.416 (4) p-value: p=0.7364 (from calculator t-test) (5) Conclusion: compare p-value to ∝ Is 0.7364 ≤ 0.10? No. Do not reject. (6) There is not sufficient evidence that the average number of daily t.v. viewing hours has changed from that in 2002 (average of 4.66 hours daily).
A snack food company produces a 454 g bag of pretzels and insists that the mean net weight of the bags is 454 g. As part of its program, the quality assurance department periodically performs a hypothesis test to decide whether the packaging machine is working properly, that is, to decide whether the mean net weight of all bags packaged is 454 g. (1) Determine the null hypothesis for the hypothesis test. (2) Determine the alternative hypothesis for the hypothesis test. (3) Classify the hypothesis test as two tailed, left tailed, or right tailed.
(1) H₀: µ = 454 g (2) H₁: µ ≠ 454 g (3) two tailed
A hot tub manufacturer advertises that with its heating equipment a temperature of 100 degrees F can be achieved in at most 15 minutes. A random sample of 20 tubs is selected and the time needed to reach 100 degrees is determined for each tub. The sample mean is 16 minutes with a standard deviation of 1 minute. Does this information cast doubt on the company's claim? Assume ∝ = 0.01.
(1) H₀: µ ≤ 15 H₁: µ > 15 (2) ∝ = 0.01 (3) Test statistic: t = 4.47 (4) P-value = .000013 (using t-test on calculator) (5) Compare p-value to ∝ .000013 ≤ 0.01 Reject H₀ (6) There is enough evidence to suggest the average time for a hot tub to reach 100 degrees F is more than 15 minutes (p=.000013, ∝ = 0.01).
Decide in the following situations whether the z-test is an appropriate method for conducting the hypothesis test for a population mean: (a) no outliers, distribution highly skewed, sample size 24, (b) no outliers, mildly skewed, sample size 70
(a) not appropriate (b) appropriate
For which of the following p-values would the null hypothesis be rejected at a level of ∝ = 0.05: (a) .001 (b) .021 (c) .078 (d) .047 (e) .148
(a) reject (b) reject (c) do not reject (d) reject (e) do not reject
Calculation of critical values (for hypothesis test for one population when ϑ is known)
+/- Z (∝/2) - two tailed test - Z ∝ - left tailed test Z∝ - right tailed test
Using the test statistic formula for the z-score, determine the required critical value(s) for a left-tailed test with ∝ = 0.05.
-1.645
Using the test statistic formula for the z-score, determine the required critical value(s) for a two-tailed test with ∝ = 0.05.
-1.96 and +1.96
Using the test statistic formula for the z-score, determine the required critical value(s) for a right-tailed test with ∝ = 0.05.
1.645
What is the appropriate t-value for the following confidence level and sample size: confidence level 90%, n=12
1.796
What is the appropriate t-value for the following confidence level and sample size: confidence level 95%, n=17
2.120
What is the appropriate t-value for the following confidence level and sample size: confidence level 99%, n=24
2.807
A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate with 95% confidence to within 0.1 lb, the average force required to break the binding? Assume ϑ is known to be 0.8 lb.
246
A confidence interval for a population mean has a margin of error of 3.4 Determine the length of the confidence interval.
6.8
What mathematical signs are allowed in the null hypothesis?
=, ≤, ≥
Explain the meaning of the term hypothesis as used in inferential statistics.
A hypothesis is a statement that something is true.
one-tailed test
A hypothesis test that is either left-tailed or right-tailed.
How does a large test statistic relate to the area in the tail?
A large test statistic means that there is a smaller area in the tail.
How does a small test statistic relate to the area in the tail?
A small test statistic means that there is a larger area in the tail.
Does this pair comply with the rules for setting up hypotheses? If not explain why. H₀: µ = 123; H₁: µ < 123
Complies
Does this pair comply with the rules for setting up hypotheses? If not explain why. H₀: µ = 50; H₁: µ ≠ 50
Complies
Suppose a CEO of a company wants to determine whether the average amount of wasted time during an 8-hour day for employees at the company is less than 120 minutes. A random sample of 10 employees gave these results: 108, 131, 112, 113, 117, 113, 130, 105, 111, 128 Assume ϑ = 9. Do these data provide evidence that the mean wasted time for this company is less than 120 minutes?
Conclusion: Do not reject H₀ Interpretation: There is not enough evidence to conclude that the average amount of wasted time at the company is less than 120 minutes.
Does this pair comply with the rules for setting up hypotheses? If not, explain why. H₀: µ = 15; H₁: µ = 15
Does not comply. H₁ must be stated as ≠ 15, <15, or > 15.
Does this pair comply with the rules for setting up hypotheses? If not explain why. H₀: µ = 123; H₁: µ = 125
Does not comply. H₁ must use the same number as H₀ and cannot contain the equal sign.
Does this pair comply with the rules for setting up hypotheses? If not explain why. H₀: µ = 10; H₁: µ > 12
Does not comply. H₁ must use the same number as H₀, the null hypothesis.
True or False: The confidence interval can be obtained if you know only the margin of error.
False
What is the relationship between Type I and Type II Error probabilities?
For a fixed sample size, the smaller we specify the significance level, ∝, the larger will be the probability, β, of not rejecting a false null hypothesis.
Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The Czech and Slovak governments have set a safety limit for cadmium in dry vegetables at 0.5 parts per million (ppm). A random sample of the edible mushroom Boletus pinicola with the resulting data: 0.24, 0.59, 0.62, 0.16, 0.77, 1.33, 0.92, 0.19, 0.33, 0.25, 0.59, 0.32 . At the 5% significance level, do the data provide sufficient evidence to conclude that the mean cadmium level in Boletus pinicola mushrooms is greater than the government's recommended limit of 0.5 ppm? Assume that the population standard deviation of cadmium levels in Boletus pinicola mushrooms is 0.37 ppm. (Note: The sum of the data is 6.31 ppm).
Given: significance level 0.05 ϑ = 0.37 H₀: µ = 0.5 ppm H₁: µ > 0.5 ppm Test statistic: z=0.24 Critical value: 1.645 Conclusion: 0.24 is in the non-rejection region Interpretation: There is not enough evidence to conclude that the mean level of cadmium in Boletus pinicola mushrooms is greater than 0.5 ppm.
The mean charitable contribution per household in the U.S. in 2000 is $1623. A researcher claims that the level of giving has changed since then. State the null and the alternative hypotheses.
H₀: µ = $1623 H₁: µ ≠ $1623
Researchers have postulated that because of differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170. Let µ represent the true mean blood cholesterol level for Japanese children. What hypothesis should the researchers test? Give the null and alternative hypotheses.
H₀: µ = 170 (the null hypothesis is the hypothesis to be tested) H₁: µ < 170
Federal law requires that a jar of peanut butter labeled 32 oz. must contain at least 32 oz. A consumer advocate feels that a certain manufacturer is shorting customers by underfilling jars so that the mean content is less than 32 oz. State the null and alternative hypotheses.
H₀: µ ≥ 32 H₁: µ < 32 left tailed test
Possible conclusions for a hypothesis test
If the null hypotheses is rejected, we conclude that the alternative hypothesis is true. If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis.
two-tailed test
If the primary concern is deciding whether a population mean, µ, is different from a specified value µ₀, we express the alternative hypothesis as: H₁: µ ≠ µ₀
right-tailed test
If the primary concern is deciding whether a population mean, µ, is greater than a specified value µ₀, we express the alternative hypothesis as: H₁: µ > µ₀
left-tailed test
If the primary concern is deciding whether a population mean, µ, is less than a specified value µ₀, we express the alternative hypothesis as: H₁: µ < µ₀
