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Air Temperature How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds by Wirth and Young (Random House) claims that the air in the crown should be an average of \(100^{\circ} \mathrm{C}\) for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly \(100^{\circ} \mathrm{C}\). What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 56 readings (for a balloon in equilibrium) gave a mean temperature of \(\bar{x}=97^{\circ} \mathrm{C} .\) For this balloon, \(\sigma \approx 17^{\circ} \mathrm{C}\) (a) Compute a \(95 \%\) confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (b) Interpretation If the average temperature in the crown of the balloon goes above the high end of your confidence interval, do you expect that the balloon will go up or down? Explain.

Step by step solution

01

Introduction to the Confidence Interval

We are asked to compute a 95% confidence interval for the average temperature at which the balloon is in equilibrium. To do this, we will use the formula for the confidence interval of a normal distribution since the sample size is large (56 readings): \[ \mu = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)\]where \(\bar{x}\) is the sample mean, \(\sigma\) is the standard deviation, \(n\) is the sample size, and \(Z\) is the Z-score corresponding to the desired confidence level.

02

Determine the Z-score for 95% Confidence Level

For a 95% confidence level, the critical value \(Z\) is often found in statistical tables or calculated by statistical software. For a 95% confidence level, \(Z \approx 1.96\).

03

Calculate the Standard Error

We calculate the standard error using the formula:\[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}}\]Using the given values, \(\sigma = 17^{\circ}C\) and \(n = 56\), we have:\[ \text{Standard Error} = \frac{17}{\sqrt{56}} \approx 2.27\]

04

Compute the Confidence Interval

Now substitute \(\bar{x} = 97\), \( Z = 1.96\), and the standard error calculated in the previous step into the confidence interval formula:\[ \mu = 97 \pm 1.96 \times 2.27\]Thus the confidence interval is:\[ \mu = 97 \pm 4.45\]\[ \mu \approx (92.55, 101.45)\]

05

Interpretation of Confidence Interval

The 95% confidence interval for the average temperature is approximately \((92.55, 101.45)\) degrees Celsius. This means we are 95% confident that the true average temperature for the balloon's equilibrium is within this range.

06

Analysis of Balloon's Reaction to Temperature Changes

If the balloon's crown temperature goes above the high end of the confidence interval (above \(101.45^{\circ} C\)), it is likely that the balloon will tend to go up, as higher temperatures create more lift by reducing air density inside the balloon.

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Key Concepts

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

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how spread out the data points are around the mean of the dataset. In the context of the air temperature in a hot air balloon, a standard deviation of \(17^{\circ}C\) means that most of the temperature readings are within \(17^{\circ}C\) of the average (mean) temperature collected in the sample set. This gives us insight into the variability we can expect in our measurements.

The formula for standard deviation, \(\sigma\), is:

• \[ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \]

where \(x_i\) represents each data point, \(\bar{x}\) is the sample mean, and \(n\) is the number of observations. A smaller standard deviation implies that the data points are closer to the mean, indicating a more consistent set of measurements.

In this exercise, the stated standard deviation helps us calculate the confidence interval, which in turn, helps us confirm a safe operational temperature range for the balloon.

Normal Distribution

A normal distribution, often referred to as the "bell curve," is a statistical distribution where most observed data points cluster around the mean and the probabilities of values taper symmetrically as they move away from the mean. It is important in this exercise because the concept of a confidence interval relies on the assumption that the data follows a normal distribution, especially when the sample size is large enough (in this case, 56 readings).

Some key characteristics of a normal distribution include:

• It is symmetric around its mean.
• The mean, median, and mode are all equal.
• Approximately 68% of data falls within one standard deviation from the mean.
• About 95% of data falls within two standard deviations from the mean.
• Almost all (99.7%) of data falls within three standard deviations from the mean.

This understanding of normal distribution allows engineers and scientists to make predictions and informed decisions based on data, as done in this example with temperature data from the balloon.

Z-score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. For instance, if a Z-score is 1, it indicates that the data point is one standard deviation above the mean.

Z-scores are particularly useful in this exercise because they help in determining the confidence level of our estimates. A 95% confidence level uses a Z-score of approximately 1.96, which corresponds to the range within which we expect 95% of the data values to fall in a normal distribution. This critical value ensures that the calculation of the confidence interval captures the true mean with the desired level of confidence.

The Z-score formula is:

• \[ Z = \frac{(X - \mu)}{\sigma} \]

where \(X\) is the value we are examining, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. By understanding and utilizing Z-scores, researchers can interpret how extreme a given observation is, in relation to an average.

Sample Mean

The sample mean, denoted as \(\bar{x}\), is the average of all data points in a sample set. It provides an estimate of the population mean when it is impractical to measure every single data point in a large dataset. In this scenario, you have a sample mean of \(97^{\circ}C\) calculated from 56 readings of the temperature inside the balloon.

The formula for the sample mean is:

• \[ \bar{x} = \frac{\sum x_i}{n} \]

where \(x_i\) are the individual data points and \(n\) is the sample size. The sample mean gives an insight into the central tendency of the dataset, and it forms the basis for constructing confidence intervals.

In practice, knowing the sample mean allows for interpreting and predicting the behavior of the temperature inside the balloon, essential for maintaining equilibrium and ensuring safe operation. It acts as a pivotal reference point to compare with new temperature readings during flight, guiding necessary adjustments.