Statistic: Chapter 22
Why did the data give "sufficient evidence" to conclude Ha was correct?
. Because P-value was less than α.
Suppose we are testing H0: µ = 50 versus Ha: µ > 50. What is the value of µ0?
50
What should you conclude when P-value > α?
Conclude "insufficient evidence to believe Ha."
What should you conclude when P-value < α?
Conclude Ha is correct.
What is the symbol for the null hypothesis?
H0
What is the probability of getting a sample mean of 16.1 ounces or higher IF µ were 16 ounces?
High (greater than 0.10)
What does µ represent?
The mean weight of all bottles of chocolate milk filled by the machine
For P-value = 0.2358 and α = 0.05, can we conclude Ha is correct?
no
If P-value > α, our results are _____.
not statistically significant at level α. Results that are not significant are results that are likely to happen by chance if the null hypothesis were true.
If the symbol < is found in the alternative hypothesis, we declare the test to be ______-sided.
one < is only one direction (less than the claimed value), so it is a one-sided test.
A test of significance tests a claim about a _______________.
parameter
What does "P" represent in "P-value?"
probability
When conducting a statistical test of significance, the standard score that is computed based on the sample data is commonly referred to as the _________________.
test statistic
Which of the following is a good synonym for "significant?"
too large
What symbol should always be used in the null and alternative hypotheses when testing the mean?
μ Hypotheses are about parameters and the parameter for the mean is μ.
In a test of significance we want to find evidence against the _____.
null hypothesis The null hypothesis is the hypothesis that we are trying to find evidence against.
What is the appropriate alternative hypothesis for testing, "Is the mean for all BYU students less than 6?"
Ha: μ < 6
What is the appropriate alternative hypothesis for testing, "Has the training raised the mean above 483?"
Ha: μ > 483
What is the appropriate alternative hypothesis for testing, "Does the mean systolic blood pressure of all executives differ from 128?"
Ha: μ ≠ 128
Are the chances of an all male jury high or low given that 50% of the population is male?
Low
Because the research question is "Is the mean less than 76.5," what symbol should we use in Ha?
<
For what P-values do we reject H0?
Low P-values
For P-value = 0.2358 and α = 0.05, should we reject H0?
. No
True or false: If the data were not collected with a simple random sample, the P-value computed from these data would be inaccurate.
. True
True or false: Statistically speaking, we can conclude "he loves me" because the probability of sending roses is high when he loves me.
.False
The farther the value of ¯¯¯x is from the claimed value of µ on the sampling distribution of ¯¯¯x curve, the smaller the P-value.
.True
For a certain hypothesis test the observation is 1.26, the mean is 1.23 with a standard deviation of 0.03. Find the test statistics.
1
If we wish to decide whether the mean is "less than" 16, which inequality symbol do we use in Ha?
<
If we wish to decide whether the mean is "greater than" 16, which inequality symbol do we use in Ha?
>
Which symbol should we use in Ha to test the research question, "Was the mean greater than 3?"
>
Why is testing H0: μ = 75.6 versus Ha: µ < 75.6 a "one-sided" test?
Because "<" in Ha is one-directional.
Can we conclude that the mean weight of the bottles is not 16 ounces?
No, because x= 16.1 (or higher) is likely to happen if µ were 16 ounces.
For P-value = 0.2758 and α = 0.05, should we reject H0?
No
If α = 0.10 and P-value = 0.236, can we reject H0?
No
If α= 0.10 and P-value = 0.236, are the results statistically significant?
No
Which hypothesis do we usually wish to find evidence "against?"
The null hypothesis
A sample result is statistically significant at the 5% level if:
The result would occur by chance no more than 5% of the time.
Which of the following is true about the P-value?
The smaller the P-value is, the stronger the evidence provided by the data against the null hypothesis.
How do we know which tail of the sampling distribution of ¯¯¯x contains the P-value of a test?
The tail containing P-value is specified by the direction in Ha.
A significance test looks for evidence against the null hypothesis and in favor of the alternative hypothesis.
True
The null hypothesis is a statement of "no difference" or "no change."
True
True or false: One should never use the symbol x in H0 or Ha.
True
We assume H0 is correct when computing P-value.
True
α is the symbol for level of significance.
True
What is the probability of getting a sample mean of 16.3 ounces or higher IF µ were 16 ounces?
Very low (smaller than 0.10)
When is an observed effect statistically significant?
When it is "too large" to be due to chance.
When do we reject H0?
Whenever P-value is less than α
Suppose we are testing a claim that μ = 80 and that the standard deviation of the sampling distribution of equals 4. Which of the following sample means gives the most evidence against the claim?
x¯= 89 This value is furthest from the claimed mean of 80, so is least likely to happen.
If α= 0.05 and P-value = 0.017, are the results statistically significant?
yes
If α= 0.05 and P-value = 0.017, can we reject H0?
yes
Which symbol should we use in Ha to test the research question, "Has the mean changed from 11.50?"
≠
Why can we conclude that the mean weight of the bottles is not 16 ounces?
16.3 (or higher) would rarely happen if µ were 16 ounces.
Since z = 3.86 is unlikely, do we have evidence for H0 or against H0?
Against H0
Why is testing H0: µ = 3 versus Ha: µ > 3 a "one-sided" test?
Because ">" in Ha is one-directional.
Why is testing H0: μ = 11.50 versus Ha: µ ≠11.50 a "two-sided" test?
Because "≠" in Ha is two-directional.
Why is a test statistic of z = 1.29 "not statistically significant?"
Because 95% of all z-scores are between -2 and +2 and 1.29 is between these two values so it is likely.
Why is a test statistic value of z = 3.86 "statistically significant?"
Because 95% of all z-scores are between -2 and +2 and 3.86 is outside these two values so it is unlikely.
Why did the data give "insufficient evidence" to conclude Ha was correct?
Because P-value was greater than α.
Why is P-value the area in the upper tail of the sampling distribution of ¯¯¯x
Because we have "greater than" (>) in Ha.
Why is P-value the area in the lower tail of the sampling distribution of ¯¯¯x
Because we have "less than" (<) in Ha.
Why is P-value the area in both tails of the sampling distribution of ¯¯¯x
Because we have "not equal to" (≠) in Ha.
When computing the test statistic, z=¯¯¯x−μoσ√n how do we assume the parameter claim were true?
By using µ0 in the formula
P-value is the probability that the null hypothesis were true.
False
The claim being tested in a hypothesis test is called the alternative hypothesis.
False
True or false: P-value is the probability that H0 is correct.
False
True or false: The closer the observed statistic value is from the claimed parameter value, the more unlikely the statistic value is.
False
True or false: The promotion rate for Hispanics was about the same as the promotion rate for all FBI agents.
False
You have available data showing that 81% of all eligible students in Connecticut took the SAT during the 1994-1995 school year. You are interested in testing whether the proportion of eligible students in Connecticut that plan to take the SAT during 2010-2011 school year is significantly different than the proportion that took the SAT during the 1994-1995 school year. Which of the following pair of hypotheses (null and alternative) is appropriate for this test?
H0: p = 0.81 versus Haa: p ≠ 0.81 This is the correct pair of hypotheses for this test.
Which of the following hypotheses are two-sided?
H0: μ = 16 versus Ha: μ ≠ 16
When is level of significance determined?
In the planning stage of a study by the researcher
To determine statistical significance, what do we compute a probability on?
On getting a statistic value as far or farther from the observed statistic value
When computing P-value of 0.0018, what did we compute the probability on?
On getting ¯¯¯x = 5.2 or lower
When computing P-value, what did we compute the probability on?
On getting ¯¯¯x= 499.1 or greater
You have available data showing that 81% of all eligible students in Connecticut took the SAT during the 1994-1995 school year. You are interested in testing whether the proportion of eligible students in Connecticut that plan to take the SAT during 2010-2011 school year is significantly different than the proportion that took the SAT during the 1994-1995 school year. You take an SRS of size 1000 of Connecticut students eligible to take the SAT and find that 85% plan to take the SAT during the 2010-2011 school year. What is the region corresponding to the P-value for the test of hypothesis?
Two-sided The P-value will correspond to the region in both tails because the alternate hypothesis is that the proportion is significantly different than 81%.
If we wish to decide whether the mean is "changed from" 16, which inequality symbol do we use in Ha?
equal cross
When do we conclude that a claim about a parameter value is incorrect?
When the probability of getting a statistic value is unlikely if the claim were to be correct.
When do we have good evidence that a claim about a parameter is not true?
When the probability of getting the observed value of the statistic assuming the claim were true is really small.
When are results statistically significant?
Whenever P-value is small
For P-value = 0.0018 and α = 0.10, can we conclude Ha is correct?
Yes
For P-value = 0.0018 and α = 0.10, should we reject H0?
Yes
A group of researchers wanted to know if there was a difference in average yearly income taxes paid between residents of two very large cities in the Midwestern United States. The average for the first city was $6,505 and for the second city, it was $6,511. The difference provided a P-value of 0.0007. Were these results statistically significant at the α = 0.05 level?
Yes, because the P-value is small. 0.0007 is a small P-value. We would reject a null hypothesis with a P-value like this for any typical α level.
We use the _____ distribution to compute the P-value for a test of hypothesis for p.
Z The type of test is a z, so we use the standard Normal distribution to find the P-value.
0.05 is _____ value for α.
a reasonable 0.05 is a reasonable value for α.
The statement we hope or suspect is true instead of the null hypothesis is called the ______________.
alternative hypothesis
Zesty Pizza claims that their mean delivery time is 30 minutes. Because you had to wait so long for your pizza, you decide to test whether the average delivery time is longer than 30 minutes. A random sample of 15 deliveries had a mean delivery time of x¯= 33.5 minutes. You should declare statistical significance only if there is a small probability of _____.
observing a sample mean of 33.5 minutes or greater when μ = 30 minutes. Because you suspect that the mean is greater than 30 minutes, the probability must be about observing a sample mean of 33.5 minutes or more.
Fill in the blank: An observed effect that would ________ happen if a claim about a parameter value were true is good evidence that the claim is not true.
rarely
If the P-value < α, the results of a test are statistically _____.
significant We reject the null hypothesis when P-value < α. "Statistically significant" means we would reject the null hypothesis.