Question

Snow avalanches can be a real problem for travelers in the western United States and Canada. A very common type of avalanche is called the slab avalanche. These have been studied extensively by David McClung, a professor of civil engineering at the University of British Columbia. Suppose slab avalanches studied in a region of Canada had an average thickness of μ = 67 cm. The ski patrol at Vail, Colorado, is studying slab avalanches in its region. A random sample of avalanches in spring gave the following thicknesses (in cm). 59 51 76 38 65 54 49 62 68 55 64 67 63 74 65 79 (i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) x = cm s = cm (ii) Assume the slab thickness has an approximately normal distribution. Use a 1% level of significance to test the claim that the mean slab thickness in the Vail region is different from that in the region of Canada. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 67; H1: μ > 67 H0: μ = 67; H1: μ < 67 H0: μ ≠ 67; H1: μ = 67 H0: μ = 67; H1: μ ≠ 67 H0: μ < 67; H1: μ = 67 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The Student's t, since we assume that x has a normal distribution and σ is known. The standard normal, since we assume that x has a normal distribution and σ is known. The Student's t, since we assume that x has a normal distribution and σ is unknown. The standard normal, since we assume that x has a normal distribution 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 > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (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 the mean slab thickness in the Vail region is different from that in the region of Canada. There is insufficient evidence at the 0.01 level to conclude that the mean slab thickness in the Vail region is different from that in the region of Canada.

Answer 1

HYPOTHESIS TEST-

Suppose, random variable X denotes thickness (in cm) of slab avalanches in Vail region during spring.

We have sample values. But we do not know population standard deviation (or variance). So, we have to perform one sample t-test.

We have to test for null hypothesis

against the alternative hypothesis

Our test statistic is given by

Here,

Sample size

Sample mean is given by

Sample standard deviation is given by

Degrees of freedom

[Using R-code 'pt(-1.948872,15)+1-pt(1.948872,15)']

Level of significance

We reject our null hypothesis if

Here, we observe that

So, we cannot reject our null hypothesis.

Hence, based on the given data we can conclude that there is no significant evidence that mean thickness differs from 67 cm.

ANSWERS-

(i)

Sample mean is 61.81 cm.

Standard deviation is 10.65 cm.

(ii)

(a)

Level of significance is

Null hypothesis is

Alternative hypothesis is

(b)

We shall use the Student's t, since we assume that x has a normal distribution and σ is unknown.

Value of test statistic is .

(c)

From our hypothesis test we get, 0.050 < P-value < 0.100.

Area corresponding to P-value in our sampling distribution is as follows.

(d)

From our answers in (a) to (c) we have, at the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e)

We conclude that, there is insufficient evidence at the 0.01 level to conclude that the mean slab thickness in the Vail region is different from that in the region of Canada.