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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 had an average thickness of \(\mu=67 \mathrm{~cm}\) (Source: Avalancbe Handbook, by D. McClung and 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 \(\mathrm{cm}\) ): \(\begin{array}{lllllll}59 & 51 & 76 & 38 & 65 & 54 & 49\end{array}\) 62 \(\begin{array}{llllllll}68 & 55 & 64 & 67 & 63 & 74 & 65 & 79\end{array}\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x} \approx 61.8 \mathrm{~cm}\) and \(s \approx 10.6 \mathrm{~cm} .\) ii. Assume the slab thickness has an approximately normal distribution. Use \(1 \%\) 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

Understand the Problem

We are asked to verify the calculated sample mean (\(\bar{x}\)) and standard deviation (\(s\)) for given slab avalanche thickness data. Then, we need to perform a hypothesis test to see if the mean thickness in Vail differs from that in Canada (\(\mu = 67\,\mathrm{cm}\)).

02

Verify the Mean and Standard Deviation

Compute the mean (\(\bar{x}\)) by adding all data points and dividing by the number of data points: \(\bar{x} = \frac{59+51+76+38+65+54+49+62+68+55+64+67+63+74+65+79}{16} \approx 61.8\,\mathrm{cm}\). For \(s\), use the standard deviation formula: \[ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2} \approx 10.6\,\mathrm{cm}\]."

03

State the Hypotheses

Null hypothesis \(H_0: \mu = 67\,\mathrm{cm}\). Alternative hypothesis \(H_a: \mu eq 67\,\mathrm{cm}\). These hypotheses test whether the mean thickness differs from 67 cm.

04

Conduct the Hypothesis Test

Use a t-test for one sample mean. Calculate the t-statistic using: \[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} = \frac{61.8 - 67}{10.6/\sqrt{16}} \approx -1.962 \]. Determine the critical t-value for a 1% significance level and 15 degrees of freedom from the t-distribution table.

05

Decision Based on Critical Value

If the absolute t-statistic value is greater than the critical t-value, reject the null hypothesis. For a 1% significance level and 15 degrees of freedom, \(t_{critical} \approx 2.947\). Here, \(|-1.962| < 2.947\), so we do not reject the null hypothesis.

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Key Concepts

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

Slab Avalanche

Slab avalanches are one of the most common and dangerous types of snow avalanches. They occur when a relatively cohesive plate of snow, supported on a weaker layer beneath it, breaks away and slides down a slope. This type of avalanche is seen frequently in the western parts of the United States and Canada. These can be triggered naturally or by human activities, such as skiing.
The specific area affected by a slab avalanche and its thickness are important to study as they help in understanding the risk and planning safety measures. Researchers like David McClung have extensively researched slab avalanches to predict their occurrence and mitigate their effects. In our context, knowing the average slab thickness, such as the 67 cm found in Canadian studies, serves as a benchmark for comparison with other regions like Vail, Colorado.

Sample Mean

The sample mean, denoted as \( \bar{x} \), is a statistical measure that represents the average value in a sample dataset. It is computed by summing all the observed values and dividing by the number of observations. In the case of slab avalanche data from Vail, Colorado, the ski patrol calculated the sample mean thickness of the avalanches to be approximately 61.8 cm.
Calculating the sample mean provides a central tendency and is crucial for comparing different datasets. It serves as a crucial step in statistical analysis, such as hypothesis testing, making it easier to determine if the sample matches expectations drawn from a known population, like the observed avalanche thicknesses in Canada.

Standard Deviation

Standard deviation is another important statistical measure that quantifies the amount of variation or dispersion in a set of values. For slab avalanches, knowing the standard deviation helps in understanding how much the thickness of different avalanches deviates from the mean. In our case, the standard deviation of the sample from Vail was found to be approximately 10.6 cm.
The formula for standard deviation is:

• Calculate the mean of the dataset.
• Find the squared difference between each data point and the mean.
• Average those squared differences.
• Take the square root of that average.

This process provides insights into the reliability of the mean as a representation of the data, showing how tightly or loosely the data points cluster around the mean. A larger standard deviation suggests more variation among the thicknesses measured.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing, representing a default position that there is no effect or difference. In the context of our slab avalanche study, the null hypothesis presumes that the mean slab thickness in Vail is the same as that in Canada, or 67 cm. It is generally denoted as \( H_0 \).
Formulating the null hypothesis involves stating expectations based on existing knowledge or previous findings. For the ski patrol at Vail, rejecting or failing to reject the null hypothesis helps make data-driven conclusions about whether local avalanche conditions differ significantly from those in Canada. If statistical tests indicate the alternate hypothesis, \( H_a \), is more likely, then the null hypothesis is rejected, signaling that observed differences are statistically significant.

T-test

A t-test is a statistical test used to determine if there is a significant difference between the means of two groups. In this exercise, a t-test helps determine whether the mean thickness of slab avalanches in Vail significantly differs from that in Canada. The calculation involves:

• The sample mean \( \bar{x} \).
• The known population mean \( \mu \) (67 cm here).
• The sample standard deviation \( s \).
• The number of samples \( n \).

The t-statistic is computed using the formula:\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]In this study, the t-statistic was around -1.962. For a significance level of 1%, with 15 degrees of freedom, the critical t-value is approximately 2.947. Since the calculated t-value does not exceed this critical value, the null hypothesis fails to be rejected, indicating no significant difference in slab thickness between Vail and Canada.