Question

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 = 55 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 = 41 stemless projectile points showed the mean length to be x₂ = 2.70 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?

difference of meanspaired difference    single proportiondifference of proportionssingle mean

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

What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with unknown population standard deviations.The Student's t. We assume that both population distributions are approximately normal with known population standard deviations.    The Student's t. We assume that both population distributions are approximately normal with unknown population standard deviations.The standard normal. We assume that both population distributions are approximately normal with known population standard deviations.

What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to three decimal places.)


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.

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.

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.    

Answer 1

What are we testing in this problem?

difference of means

What is the level of significance?

5% or 0.05

State the null and alternate hypotheses.

H0: μ1 = μ2; H1: μ1 ≠ μ2

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 unknown population standard deviations.

What is the value of the sample test statistic?

n1 = 55 , = 3.00 , s1 = 0.80

n2 = 41 , = 2.70 , s2 = 0.90

1 = 2 = [Equal variance]

N( 1 , ^2 / 55)

N( 2 , ^2 / 41)

( - ) N( 1 - 2 , ^2 / 55 + ^2 / 41)

population standard deviation is estimated by pooled standard deviation.

= Sp = pooled standard deviation

Pooled variance,

Sp2 = ( (n1-1) S1 2 + (n2-1) S2 2 ) / (n1+n2-2)

=( 54* .8*.8 + 40 *.9 * .9 ) / 94

= .7123

Sp = .844

So, ( ( - ) - ( 1 - 2 ) ) / Sp t n1+n2 -2

This is two smaple two tail t test.

Test statistic , t = ( - ) / Sp = ( 3- 2.7) / .844 = 1.722 (sample statistic)

Estimate the P-value.

t /2=0.025 , 94 = 1.984

t /2=0.05 , 94 = 1.660

our test staistic lie in between these 2 value , so 0.050 < P-value < 0.1000

Sketch the sampling distribution and show the area corresponding to the P-value.

Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

Since p value is grater than 0.05 , At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (last option).

Interpret your conclusion in the context of the application.

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.   

I answered all the subparts.

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