The Wind Mountain archaeological site is in southwest New
Mexico. Prehistoric Native Americans called Anasazi once lived and
hunted small game in this region. A stemmed projectile point is an
arrowhead that has a notch on each side of the base. Both stemmed
and stemless projectile points were found at the Wind Mountain
site. A random sample of n1 = 60 stemmed
projectile points showed the mean length to be
x1 = 3.00 cm, with sample standard deviation
s1 = 0.80 cm. Another random sample of
n2 = 46 stemless projectile points showed the
mean length to be x2 = 2.67 cm, with
s2 = 0.90 cm. Do these data indicate a
difference (either way) in the population mean length of the two
types of projectile points? Use a 5% level of significance.
What are we testing in this problem?
single meansingle proportion paired differencedifference of proportionsdifference of means
(a) What is the level of significance?
State the null and alternate hypotheses.
H0: μ1 = μ2; H1: μ1 ≠ μ2H0: μ1 = μ2; H1: μ1 < μ2 H0: μ1 = μ2; H1: μ1 > μ2H0: μ1 ≠ μ2; H1: μ1 = μ2
(b) What sampling distribution will you use? What assumptions are
you making?
The Student's t. We assume that both population distributions are approximately normal with known standard deviations.The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
What is the value of the sample test statistic? (Test the
difference μ1 − μ2. Round
your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value > 0.5000.250 < P-value < 0.500 0.100 < P-value < 0.2500.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010
Sketch the sampling distribution and show the area corresponding to
the P-value.
(d) Based on your answers in parts (a) to (c), will you reject or
fail to reject the null hypothesis? Are the data statistically
significant at level α?
At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
(e) Interpret your conclusion in the context of the
application.
There is sufficient evidence at the 0.05 level to conclude that there is a difference in mean length of the two types of projectile points.There is insufficient evidence at the 0.05 level to conclude that there is a difference in mean length of the two types of projectile points.
difference of means
a)
level of significance =0.05
H0: μ1 = μ2; H1: μ1 ≠ μ2
b)
the Student's t. We assume that both population distributions are approximately normal with unknown standard deviations
std error =√(S21/n1+S22/n2)= | 0.1682 | |
test stat t =(x1-x2-Δo)/Se = | 1.963 |
0.050 < P-value < 0.10
d)
At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
e)
There is insufficient evidence at the 0.05 level to conclude that there is a difference in mean length of the two types of projectile points.
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