Download Statistics – Hypothesis Testing (Ch 9) – Review. KEY

Statistics – Hypothesis Testing (Ch 9) – Review.
KEY
You should complete this review, with work shown, to be eligible to test. You may use a 3x5 card of notes on the test.
All probability answers must be either a completely reduced fraction, a decimal rounded to 4 places or a percent rounded to 2
places. Unless instructed otherwise, round any other response to 4 decimal places.
1.
State the requirements that must be satisfied in order to test a claim about a population mean when the population standard
deviation is known. What type of distribution is used? Normal distribution (z-scores)
1. Sample is randomly selected. 2. Either 𝒏 ≥ 𝟑𝟎 or the sample is normally distributed with no outliers.
2.
State the requirements that must be satisfied in order to test a claim about a population mean when the population standard
deviation is not known. What type of distribution is used? t-distribution (t-scores)
1. Sample is randomly selected. 2. Either 𝒏 ≥ 𝟑𝟎 or the sample is normally distributed with no outliers.
3.
State the requirements that must be satisfied in order to test a claim about a population proportion. What type of distribution
is used? Normal distribution (z-scores)
1. Sample is randomly selected. 2. Sample must be no more than 5% of the population. 3. 𝒏𝒑𝒐 (𝟏 − 𝒑𝒐 ) ≥ 𝟏𝟎
4.
State the requirements that must be satisfied in order to test a claim about a population standard deviation. What type of
distribution is used? 𝝌𝟐 distribution (𝝌𝟐 -scores)
5.
Explain what 𝛼 represents.
The probability of making a Type I error. The probability that an innocent man is convicted. H o is rejected when it is true.
6.
Explain what 𝛽 represents.
The probability of making a Type II error. The probability that a guilty man is goes free. Ho is not rejected when it should be.
7.
Explain what the power of the test tells us about the hypothesis test.
This is the probability that when the Null (Ho) is false we reject the Null. When the power of the test is close to 1, we are not
very likely to make a Type II error (let the guilty man go free). When the power of the test is close to 0, we are likely to make
a Type II error and not likely to make a Type I error.
For 8 – 11, test the hypothesis using both the classical and the P-value approaches. Clearly state your conclusion.
8. A certain brand of apple juice is supposed to have 64 ounces of juice. Because the filling machine is not precise, the exact
amount of juice varies from bottle to bottle. The quality control manager claims that the mean amount of juice in each bottle is
not 64 ounces. She randomly samples 22 bottles of juice, measures the content, and obtains a sample mean of 63.96 ounces.
Assuming that  = 0.06 and the distribution is normal, test her claim at the  = 0.1 level of significance.
Two tailed test. Test statistic: 𝒛 = −𝟑. 𝟏𝟑, Critical Values: 𝒛 = ±𝟏. 𝟔𝟒, P-value = 0.0018. Reject the Null
There is sufficient evidence to support the claim that the average amount of juice in each bottle is not 64 ounces.
9.
In order to test H0:  = 40 versus H1:  > 40, a simple random sample of size n = 25 is obtained from a population that is known
to be normally distributed. A sample mean is found to be 42.3 and a sample standard deviation of 4.3. Test the researcher’s
claim the  = 0.1 level of significance.
Right tailed test. Test statistic: 𝒕 = 𝟐. 𝟔𝟕, Critical Value: 𝒕 = 𝟏. 𝟑𝟐, P-value = 0.0066. Reject the Null
There is sufficient evidence to support the claim that the mean is larger than 40.
10. In a recent survey, 395 out of 1012 adults aged 18 years or older said they had a gun in the house. In 1990, 47% of households
had a gun. Is there significant evidence to support the claim that the proportion of households that have a gun has decreased
since 1990 at the  = 0.05 level of significance?
̂ = −𝟓. 𝟎𝟕𝟗, Critical Value: 𝒛 = −𝟏. 𝟔𝟒, P-value = 0.00000019. Reject the Null
Left tailed test. Test statistic: 𝒑
There is sufficient evidence to support the claim that gun ownership has declined since 1990.
11. A researcher is trying to determine if the standard deviation for the amount of acidity in rain has changed from 0.2. He obtains
19 samples and finds that the standard deviation is 0.1708. Test the researcher’s claim at the  = 0.05 level of significance.
Assume that the population is normal.
Two tailed test. Test statistics: 𝝌𝟐 = 𝟏𝟑. 𝟏𝟐𝟖, Critical Value: 𝝌𝟐 = 𝟖. 𝟐𝟑𝟏, 𝝌𝟐 = 𝟑𝟏. 𝟓𝟐𝟔, P-value = 0.4321. Do not reject Ho
There is not sufficient evidence to support the claim the standard deviation is different from 0.2.
For 12 – 13, find the probability of making a Type II error and find the power of the test.
12. A researcher maintains that the average age of a woman before she has her first child is greater than the 1990 average age of
26.4 years. She obtains a simple random sample of 40 women who recently gave birth to their first child. Assume that the
population standard deviation is 6.4 years. If the researcher decides to test this hypothesis at the  = 0.05 level of significance,
the true population mean is 26.9 years.
Right tailed. Critical Value: 𝒛 = 𝟏. 𝟔𝟒, in context 28.064, 𝜷 = 𝟎. 𝟖𝟕𝟓𝟏, power of the test = 0.1249
a.
In context, what does 𝛽 tell us about this test?
We are very likely to conclude the average age of a woman is 26.4 years old before having her first child, when in fact she
is older than 26.4 years.
b.
In context, what does the power of the test tell us about this test?
We are not likely to conclude that the average age of a woman is more than 26.4 years when she is in fact 26.4 years old.
13. A school administrator claims that students whose first language learned is not English score different on the verbal portion of
the SAT exam than students whose first language is English. The mean SAT verbal score of students whose first language is
English is 516. Suppose a simple random sample of 20 students whose first language learned was not English. SAT verbal scores
are normally distributed with a population standard deviation of 109. If the researcher decides to test this hypothesis at the  =
0.05 level of significance, compute the probability of making a Type II error if the true population mean is 505.
Two tailed. Critical Values: 𝒛 = ±𝟏. 𝟗𝟔, in context 468.23 and 563.770, 𝜷 = 𝟎. 𝟗𝟐𝟔𝟒, power of the test = 0.0736
a.
In context, what does 𝛽 tell us about this test?
We are very likely to conclude that the average SAT verbal score is 516 when in fact it is not.
b.
In context, what does the power of the test tell us about this test?
We are not very likely to conclude that the average SAT verbal score is not 516 when in fact it is.