91Ó°ÊÓ

At wind speeds above 1000 centimeters per second (cm/sec), significant sand- moving events begin to occur. Wind speeds below \(1000 \mathrm{cm} / \mathrm{sec}\) deposit sand, and wind speeds above \(1000 \mathrm{cm} / \mathrm{sec}\) move sand to new locations. The cyclic nature of wind and moving sand determines the shape and location of large dunes (Reference: Hydraulic, Geologic, and Biologic Research at Great Sand Dunes National Monument and Vicinity. Colorado, Proceedings of the National Park Service Research Symposium). At a test site, the prevailing direction of the wind did not change noticeably. However, the velocity did change. Sixty wind speed readings gave an average velocity of \(\bar{x}=1075 \mathrm{cm} / \mathrm{sec} .\) Based on long-term experience, \(\sigma\) can be assumed to be \(265 \mathrm{cm} / \mathrm{sec}\). (a) Find a \(95 \%\) confidence interval for the population mean wind speed at this site. (b) Interpretation Does the confidence interval indicate that the population mean wind speed is such that the sand is always moving at this site? Explain.

Step by step solution

01

Identify the Given Data

Given data includes the sample mean wind speed, \(\bar{x} = 1075 \ \text{cm/sec}\); the population standard deviation, \(\sigma = 265 \ \text{cm/sec}\); and the number of wind speed readings, \(n = 60\). We need to construct a 95% confidence interval for the population mean \(\mu\).

02

Find the Standard Error of the Mean

The standard error (SE) of the mean is calculated using the formula: \(SE = \frac{\sigma}{\sqrt{n}}\). Substituting the values to find SE: \[ SE = \frac{265}{\sqrt{60}} \approx 34.20 \ \text{cm/sec} \]

03

Determine the Z-score for 95% Confidence

For a 95% confidence interval, the Z-score corresponding to the 95% level of confidence is approximately 1.96, which we'll use in our calculations.

04

Calculate the Confidence Interval

The confidence interval is determined by the formula: \[\bar{x} \pm Z \times SE\]Plugging in the values, we get:\[ 1075 \pm 1.96 \times 34.20 \]This calculates to:\[ 1075 \pm 66.43 \]So, the confidence interval is approximately \([1008.57, 1141.43]\ \text{cm/sec}\).

05

Interpretation of the Confidence Interval

The confidence interval \([1008.57, 1141.43]\) suggests that the population mean wind speed is likely between 1008.57 cm/sec and 1141.43 cm/sec. Since this interval is entirely above 1000 cm/sec, it indicates that the population mean wind speed is such that sand is always moving at this site.

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Key Concepts

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

Population Mean

The population mean, often represented by the symbol \( \mu \), is a measure that indicates the central tendency of a population distribution. In simpler terms, it's the average value in an entire population.
When we talk about wind speed in the context of sand dune movement, it's crucial to understand what happens on average across all measurements.

• The population mean allows us to gauge what the typical wind speed would be if we could measure every single instance of wind at the site.
• Knowing the population mean helps us predict how often and how strongly natural events, like sand movement, might occur.

However, obtaining the exact population mean isn't always feasible because it involves measuring every instance, which is why we estimate it using sample data, such as the wind speeds from our test site.

Standard Error

The standard error (SE) of the mean is an important concept when estimating a population parameter. It tells us how accurately we can expect our sample mean to represent the population mean.

• It's calculated by dividing the population standard deviation \( \sigma \) by the square root of the sample size \( n \).
• The formula is \( SE = \frac{\sigma}{\sqrt{n}} \).

This calculation shows how much the sample mean will vary from one sample to another.

• A smaller standard error indicates a more precise estimate of the population mean.
• In our example, with a standard error of 34.20 cm/sec, we get a clear window around the sample mean, showing the possible range for the actual population mean.

This provides us with insight into the reliability of our estimates and is crucial in constructing confidence intervals.

Z-score

The Z-score is a statistical measure that indicates how many standard deviations an element is from the mean. When constructing a confidence interval for a population mean, knowing the right Z-score is vital.

• A Z-score translates confidence levels into a standardized form.
• At a 95% confidence level, the Z-score is approximately 1.96. This means that 95% of the data is expected to fall within 1.96 standard deviations of the mean.

Using the Z-score in our confidence interval calculations helps us determine the margin of error for the estimate of our population mean. With a Z-score of 1.96, we accommodate most values around the mean while assessing the probability that the true mean falls within our calculated interval.
Understanding Z-scores not only assists in constructing intervals but also offers insight into data distribution and significance.

Wind Speed Analysis

Wind speed analysis in the context of sand dunes is significant as it directly impacts the movement and shaping of dunes.
Wind speeds over 1000 cm/sec result in sand moving from one place to another. This process is critical in understanding dune dynamics.

• In our analysis, we calculated a confidence interval for the population mean wind speed based on sample data.
• The interval suggests that the mean wind speed is above the threshold necessary for sand movement, indicating a continuous reshaping process.

Analyzing wind speed not only aids in predicting sand movement patterns but also provides a better understanding of the environmental forces at play.
With the data and interval calculated, we could conclude that the mean wind speed allows for constant sand transport, confirming consistent changes in dune landscapes.