91Ó°ÊÓ

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. Slab avalanches studied in Canada have an average thickness of \(\mu=67 \mathrm{cm}\) (Source: Avalanche Handbook by \(\mathrm{D.}\) McClung and \(\mathrm{P.}\) Schaerer). 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): \(\begin{array}{ccccccc}59 & 51 & 76 & 38 & 65 & 54 & 49 & 62\quad\end{array}\) \(\begin{array}{ccccc}64 & 67 & 63 & 74 & 65 & 79\quad\end{array}\) \(68 \quad 55\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x}=61.8\) and \(s \approx 10.6 \mathrm{cm}.\) 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 Canada.

Step by step solution

01

State the Hypotheses

To test the claim, we set up the null hypothesis and the alternative hypothesis. The null hypothesis states that the mean slab thickness in Vail is equal to that in Canada, while the alternative hypothesis states that there is a difference. Therefore, we have:- Null hypothesis: \(H_0: \mu = 67\)- Alternative hypothesis: \(H_a: \mu eq 67\).

02

Identify the Parameters and Sample Statistics

Given that the sample mean \(\bar{x} = 61.8\) and the sample standard deviation \(s = 10.6\). The population mean \(\mu = 67\). The sample size is \(n = 16\), as there are 16 thickness measurements.

03

Determine the Significance Level and Distribution

We are using a \(1\%\) level of significance, which means \(\alpha = 0.01\). Since the slab thickness is normally distributed and the sample size is less than 30, we will use the t-distribution. The degrees of freedom \(df = n - 1 = 15\).

04

Calculate the Test Statistic

The test statistic for a t-test is calculated using the formula:\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]Plugging in the values, we estimate:\[ t = \frac{61.8 - 67}{\frac{10.6}{\sqrt{16}}} = \frac{-5.2}{2.65} \approx -1.962 \]

05

Determine the Critical Value

For a two-tailed test at \(\alpha = 0.01\) with \(df = 15\), we check the t-distribution table for critical values. The critical t-value is approximately \( \pm 2.947 \).

06

Compare Test Statistic and Critical Value

Compare the calculated test statistic \(t \approx -1.962\) with the critical values \( \pm 2.947 \). Since \(-1.962\) lies between \(-2.947\) and \(2.947\), it falls within the acceptance region.

07

Conclusion

Since the test statistic \(-1.962\) does not fall in the rejection region defined by the critical values, we fail to reject the null hypothesis \(H_0\). There is not enough evidence to support the claim that the mean slab thickness in Vail is different from that in Canada.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the t-distribution

The t-distribution is an essential concept when dealing with statistical testing, especially for small sample sizes. In hypothesis testing, it helps determine how far a sample mean is likely to be from the actual population mean, particularly when the population standard deviation is unknown. The t-distribution is very similar to a normal distribution but has thicker tails, which means it accounts for more variability in data that may not perfectly follow the normal curve.

This makes it ideal for less extensive data sets. It's characterized by its degrees of freedom (df), which are calculated as the sample size minus one, shown as \( df = n - 1 \). The degrees of freedom affect the distribution’s shape; the fewer the degrees of freedom, the fatter the tails of the t-distribution, adding more resilience against errors from small samples.

In the context of the problem with the slab avalanches, since the sample size is 16, we use \( df = 15 \) to find the critical t-values, ensuring we understand the sample's variability.

Mean and Standard Deviation

The mean, also known as the average, and the standard deviation are fundamental concepts in statistics used to describe distributions. The mean is the sum of all sample values divided by the number of observations, providing a central point for the data. In our avalanche study, the average thickness of the sampled avalanches is \( \bar{x} = 61.8 \) cm, offering a point of comparison against the known average of 67 cm in Canada.

Meanwhile, the standard deviation measures the amount of variability or spread in a set of data. A smaller standard deviation indicates data closely clustered around the mean, while a larger one suggests a more spread out data set. In this problem, the calculated standard deviation is approximately \( s = 10.6 \) cm, demonstrating the variance in slab thickness among the samples collected in Vail.

Both mean and standard deviation are crucial for calculating the t-statistic that helps in making inferences about the population mean from the sample. They guide us in understanding and analyzing data distribution comprehensively.

Level of Significance in Hypothesis Testing

The level of significance, denoted by \( \alpha \), is a threshold set to determine the probability of making a Type I error, that is, rejecting a true null hypothesis. Common levels of significance are 5% (0.05) or 1% (0.01), indicating the maximum allowable probability of making this error in statistical conclusions.

In the avalanche example, a 1% level of significance is used, meaning we require very strong evidence against the null hypothesis for it to be rejected. By setting the significance level at \( \alpha = 0.01 \), the aim is to ensure any result concluding a difference in mean thickness is reliable and not due to random chance.

This choice directly affects the critical t-value derived from the t-distribution, dictating how extreme the test statistic must be to lead to a conclusion. It frames the decision-making in hypothesis testing, balancing the risks of errors but aiming for accuracy in findings.

Null and Alternative Hypothesis

Hypothesis testing begins with stating a null hypothesis (\( H_0 \)) and an alternative hypothesis (\( H_a \)). These provide clear statements to be tested. The null hypothesis typically reflects a position of "no effect" or "no difference," while the alternative hypothesis proposes some form of difference or effect. In the avalanche study, \( H_0: \mu = 67 \) suggests the mean slab thickness in Vail equals that in Canada.

The alternative hypothesis, \( H_a: \mu eq 67 \), proposes that there is a difference in thickness. Generating these hypotheses is a critical initial step because it defines what the study is testing.

Based on the test statistics calculated, the hypothesis testing process involves comparing these values against critical values. If the test statistic falls outside the range defined by the critical t-values, the null hypothesis is rejected. In the completion, failure to reject the null suggests there isn't enough statistical evidence to say the two means differ.