Question

A random sample of 51 adult coyotes in a region of northern Minnesota showed the average age to be x = 2.05 years, with sample standard deviation s = 0.88 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 yr

H₀: μ < 1.75 yr; H₁: μ = 1.75 yr    

H₀: μ > 1.75 yr; H₁: μ = 1.75 yr

H₀: μ = 1.75 yr; H₁: μ ≠ 1.75 yr

H₀: 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 standard normal, since the sample size is large and σ is known.

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 unknown.

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


(c) Find the P-value. (Round your answer to four decimal places.)


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

(a) What is the level of significance?

The level of significance is 0.01

The claim is that the region of northern Minnesota tend to live longer than the average of 1.75 years. So, the null and alternate hypotheses.

H₀: 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 unknown.

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

Standard error of the mean = 0.88 / = 0.1232246

Test statistic, t = (2.05 - 1.75) / 0.1232246 = 2.435

(c) Find the P-value. (Round your answer to four decimal places.)

Degree of freedom = n-1 = 51 - 1 = 50

P-value = P(t > 2.435) = 0.0093

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?

As, the p-value is less than the level of significance, we reject the null hypothesis.

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