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7.2 Hypothesis testing for the mean (sigma known)
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Explain the difference between the z-test for μ using rejection region(s) and the z-test for μ using a P-value.
In the z-test using rejection region(s), the test statistic is compared with critical values. The z-test using a P-value compares the P-value with the level of significance α.
A rejection region (or critical region) of the sampling distribution is the range of values for which the null hypothesis is not probable. A critical value z0 separates the rejection region from the nonrejection region. To use a rejection region to conduct a z-test, calculate the standardized test statistic z. If the standardized test statistic is in the rejection region, then reject H0. If the standardized test statistic is not in the rejection region, then fail to reject H0.
To use a P-value to make a conclusion in a hypothesis test, compare the P-value with α. If P ≤ α, then reject H0. If P > α, then fail to reject H0.
In hypothesis testing, does choosing between the critical value method or the P-value method affect your conclusion?
No, because both involve comparing the test statistic's probability with the level of significance
The P-value method converts the standardized test statistic to a probability (P-value) and compares this with the level of significance, whereas the critical value method converts the level of significance to a z-score and compares this with the standardized test statistic. Thus, both methods will result in the same conclusion.
Find the P-value for a left-tailed hypothesis test with a test statistic of Z=−1.15. Decide whether to reject H0 if the level of significance is α=0.05
area to the left of z= normalcdf(-10,000, -1.15,0,1)= 0.1251
p-value= 0.1251
To use a P-value to make a conclusion in a hypothesis test, compare the P-value with alphaα. If P≤ α, then reject H0. If P>α, then fail to reject H0.
Since P>α, fail to reject H0.
Find the P-value for the indicated hypothesis test with the given standardized test statistic, z. Decide whether to reject H0 for the given level of significance α.
Right-tailed test with test statistic z=1.29 and α=0.04
normalcdf(1.29,10,000,0,1)
p-value= 0.0985
Fail to reject H0 cuz p-value is higher than alpha
Find the P-value for the indicated hypothesis test with the given standardized test statistic, z. Decide whether to reject H0 for the given level of significance α.
Two-tailed test with test statistic z=−2.18 and α=0.02
Since this is a two-tailed test and the test statistic is left of center, the P-value is twice the area to the left of the test statistic
z-score-= normalcdf(-10000,-2.18,0,1)
multiplied by 2 to find p-value= 0.0292
Determine whether to reject or fail to reject H0 at the level of significance of a)α=0.07 and b) α=0.02.
H0: μ=123, Ha: μ≠123, and P=0.0396
a) Reject H0 because P<0.07
b) Fail to reject H0 because P>0.02
Find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer.
Right-tailed test, α=0.10
The critical values are z=1.28
(1-alpha= 0.9; invNorm(0.9,0,1)= 1.28
The rejection region is z >1.28
Pick the right-tailed graph
State whether the standardized test statistic z indicates that you should reject the null hypothesis. (left-tailed)
(a) z=1.208
(b) z=−1.364
(c) z=−1.467
(d) z=- 1.189
a) For z=1.208, should you reject or fail to reject the null hypothesis?
a) Fail to reject H0 because z > −1.285.
b)Reject H0 because z< −1.285.
c) Reject H0 because z<−1.285.
d)Fail to reject H0 becuase z > -1.285
A light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 750 hours. A random sample of 24 light bulbs has a mean life of 728 hours. Assume the population is normally distributed and the population standard deviation is 65 hours. At α=0.02, do you have enough evidence to reject the manufacturer's claim? This is left-tailed.
a)) Identify the null hypothesis and alternative hypothesis.
b) Identify the critical value(s).
C) Identify the standardized test statistic
d) Decide whether to reject or fail to reject the null hypothesis
(e) interpret the decision in the context of the original claim.
a) H0: mu equal than > 750 (claim)
Ha: mu < 750
b) null z: invNorm(alpha,0,1)= -2.05
c)z= -1.66 (used STAT tests menu, p. 368)
d) Fail to reject H0. There is not sufficient evidence to reject the claim that mean bulb life is at least 750 hours.
e)
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