91Ó°ÊÓ

Answer true or false. Explain your answer. A larger sample size produces a longer confidence interval for \(\mu\).

If a \(90 \%\) confidence interval for the difference of proportions contains some positive and some negative values, what can we conclude about the relationship between \(p_{1}\) and \(p_{2}\) at the \(90 \%\) confidence level?

Sam computed a \(90 \%\) confidence interval for \(\mu\) from a specific random sample of size \(n .\) He claims that at the \(90 \%\) confidence level, his confidence interval contains \(\mu .\) Is his claim correct? Explain.

Basic Computation: Confidence Interval Suppose \(x\) has a mound-shaped symmetric distribution. A random sample of size 16 has sample mean 10 and sample standard deviation 2 (a) Check Requirements Is it appropriate to use a Student's \(t\) distribution to compute a confidence interval for the population mean \(\mu\) ? Explain. (b) Find a \(90 \%\) confidence interval for \(\mu\) (c) Interpretation Explain the meaning of the confidence interval you computed.

Basic Computation: Confidence Interval A random sample of size 81 has sample mean 20 and sample standard deviation 3 (a) Check Requirements Is it appropriate to use a Student's \(t\) distribution to compute a confidence interval for the population mean \(\mu ?\) Explain. (b) Find a \(95 \%\) confidence interval for \(\mu\) (c) Interpretation Explain the meaning of the confidence interval you computed.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther (Reference: Hummingbinds by K. Long and W. Alther). A small group of 15 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is \(\bar{x}=3.15\) grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with \(\sigma=0.33\) gram. (a) Find an \(80 \%\) confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (b) What conditions are necessary for your calculations? (c) Interpret your results in the context of this problem. (d) Sample Size Find the sample size necessary for an \(80 \%\) confidence level with a maximal margin of error \(E=0.08\) for the mean weights of the hummingbirds.

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.