How hot is the air in the top (crown) of a hot air balloon?...

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.

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.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. It is constructed using the sample mean and an estimate of the standard error, and it provides a measure of the uncertainty associated with the sample estimate. This concept is fundamental in statistical inference because it quantifies the reliability of the estimation process.

Standard Error

The standard error is a measure of the variability of a sample statistic, such as the sample mean, over different samples drawn from the same population. It is calculated by dividing the population standard deviation by the square root of the sample size. The standard error is essential in determining the precision of the estimate and is used to compute the margin of error in a confidence interval.

Normal Distribution and Z Scores

When the population standard deviation is known or the sample size is large, the sampling distribution of the sample mean can be approximated by a normal distribution. Z scores, which represent the number of standard errors a value is from the mean, are used to find critical values corresponding to a given level of confidence (for example, 95%). This allows statisticians to determine the margin of error and construct the confidence interval.

Statistical Inference

Statistical inference involves using data from a sample to make estimates or test hypotheses about a population. Confidence intervals are a key tool in this area as they allow for quantifying the uncertainty in the estimates. By interpreting a confidence interval, one can assess the range within which the true population parameter is likely to fall, facilitating informed decision making based on sample data.

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