Question

A random sample of 51 adult coyotes in a region of northern Minnesota showed the average age to be x = 2.03 years, with sample standard deviation s = 0.80 years. However, it is thought that the overall population mean age of coyotes is μ = 1.75. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use α = 0.01.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ = 1.75 yr; H₁:  μ ≠ 1.75 yrH₀: μ > 1.75 yr; H₁:  μ = 1.75 yr     H₀: μ < 1.75 yr; H₁:  μ = 1.75 yrH₀: μ = 1.75 yr; H₁:  μ > 1.75 yrH₀: μ = 1.75 yr; H₁:  μ < 1.75 yr

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since the sample size is large and σ is known.The Student's t, since the sample size is large and σ is unknown.     The standard normal, since the sample size is large and σ is known.The standard normal, since the sample size is large and σ is unknown.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Estimate the P-value.

<P-value > 0.2500.100 < P-value < 0.250     0.050 < 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 = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.     At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 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 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years.There is insufficient evidence at the 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years.

Answer 1

Part a)

α = 0.01

To Test :-

H0 :- µ = 1.75
H1 :- µ > 1.75

Part b)

The Student's t, since the sample size is large and σ is unknown.  

Test Statistic :-
t = ( X̅ - µ ) / (S / √(n) )
t = ( 2.03 - 1.75 ) / ( 0.8 / √(51) )
t = 2.500

Part c)

P - value = P ( t > 2.500 ) = 0.0079

P-value < 0.010

Part d)

Reject null hypothesis if P value < α = 0.01 level of significance
P - value = 0.0079 < 0.01 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

Part e)

There is insufficient evidence at the 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years.