Problem 1 Use the Old Faithful data labelled Duration as a sample for problem 1. a. Determine the proportion of instances that the duration was less than 240 seconds. Also express the proportion as a percent. (hint: proportion is number of successes divided by total number. b. Calculate the 95% confidence interval of the proportion. What is the margin of error? c. Calculate the 90% confidence interval of the proportion. d. For a 95% confidence level and a margin of error of 3% (0.03), how many samples are needed?

Problem 2 Use the Old Faithful data labelled Duration as a sample for problem 2. Test the assertion that the mean duration time of the geyser eruption is less than 240 seconds. (hint: this is a one-sample t test for testing a claim about a mean with sigma not known -- see chap.8-4) Be sure to state: a. The mean and standard deviation of the sample b. H0 and H1 c. The value of the statistic (t) calculated d. Probability value with your conclusion, using alpha=0.05

Problem 3 Use the HWAS data for problem 3. Assume HWAS data is a Population, and use the Gender column of the data set. Take a Simple Random Sample from the Gender column of n=30. Test if the proportion of the SRS data that is female equals 0.6. (hint: this is a one-sample proportion test for testing a claim about a proportion -- see chap.8-3) Be sure to state: a. How you did the sampling, and the proportion of the SRS that is female. b. H0 and H1 c. Using alpha=0.05, the value of the statistic (z) calculated d. Probability value with your conclusion, using alpha=0.05

Problem 4 Use the HWAS data for problem 4. Assume HWAS data is a Population, and use the Age column of the data set. Take a Simple Random Sample from the Age column of n=30. a. For this SRS of 30, find the Mean, Std.Deviation, Five-Number Summary, and identify Outliers. Make histogram, and describe distribution. Description should be in sentence form and should include min, max, outliers, shape of histogram, number of peaks, etc. b. Based on the SRS, calculate the best point estimate of the population mean Age ? c. To calculate a 95% confidence level of the sample mean, determine the critical t value and the sample std.deviation. Compute the margin of error and confidence interval. (hint: See chap.7-4 on estimating population mean with sigma not known.) d. How many are needed in a sample to achieve a margin of error of 3 years at 95% confidence level? e. Using the complete Age column of the HWAS data set (the Population), make a histogram and find the mean. Compare with SRS histogram and mean from part a. Does the Population mean Age fall within the confidence interval ? Explain why it is possible for the sample and the population to be dissimilar.

We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.

We also do not have access to the data you mention.

To solve these problems, we will need to use statistical concepts and formulas. I will explain the steps to approach each problem, but please note that I am unable to directly access or analyze the data for you. You'll need to have the data or use a statistical software/tool to get the actual values needed for calculations.

Problem 1:
a. To determine the proportion of durations less than 240 seconds, you need to divide the number of durations less than 240 by the total number of instances.
- Count the number of instances where the duration is less than 240 seconds.
- Count the total number of instances.
- Divide the count of instances less than 240 by the total count to get the proportion.
- Multiply the proportion by 100 to convert it to a percentage.
b. To calculate the 95% confidence interval of the proportion, you can use the formula: proportion ± margin of error.
- The margin of error is calculated using the formula: critical value * standard error.
- The critical value depends on the desired confidence level and can be obtained from the t-distribution or z-table.
- The standard error is calculated as the square root of (proportion * (1 - proportion) / sample size).
c. Similarly, calculate the 90% confidence interval of the proportion using the same formula but with a different critical value.
d. To calculate the required sample size for a given margin of error and confidence level, you can use the formula: sample size = (z-score / margin of error)^2, where the z-score corresponds to the desired confidence level.

Problem 2:
a. Calculate the mean and standard deviation of the sample duration using the Old Faithful data.
b. State the null hypothesis (H0) as "The mean duration time of the geyser eruption is equal to or greater than 240 seconds" and the alternative hypothesis (H1) as "The mean duration time of the geyser eruption is less than 240 seconds".
c. Calculate the t-statistic using the formula: t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size)).
d. Find the probability value (p-value) associated with the calculated t-statistic and compare it to the significance level (alpha = 0.05) to make a conclusion.

Problem 3:
a. Take a simple random sample of size 30 from the Gender column of the HWAS data set. Count the number of females in the sample and divide it by the sample size to get the proportion of females.
b. State the null hypothesis (H0) as "The proportion of females in the population equals 0.6" and the alternative hypothesis (H1) as "The proportion of females in the population does not equal 0.6".
c. Calculate the z-statistic using the formula: z = (sample proportion - population proportion) / sqrt((population proportion * (1 - population proportion)) / sample size).
d. Calculate the probability value (p-value) associated with the calculated z-statistic and compare it to the significance level (alpha = 0.05) to draw a conclusion.

Problem 4:
a. Take a simple random sample of size 30 from the Age column of the HWAS data set. Calculate the mean, standard deviation, and five-number summary (minimum, first quartile, median, third quartile, maximum) of the sample. Identify any outliers.
- For the histogram, create bins and count the number of observations that fall into each bin to determine the frequency. Display the frequencies as bars with the heights corresponding to the frequencies. Provide a description of the histogram's shape, number of peaks, presence of outliers, etc.
b. The sample mean age is the calculated mean from the sample taken in part a.
c. To calculate a 95% confidence interval for the sample mean, determine the critical t-value based on the sample size and desired confidence level. Compute the margin of error as the product of the critical t-value and the sample standard deviation divided by the square root of the sample size. The confidence interval is calculated as sample mean ± margin of error.
d. Use the formula mentioned in problem 1d to determine the required sample size for a specific margin of error and confidence level.
e. Use the complete Age column of the HWAS data set (the population) to create a histogram and calculate the mean. Compare the population histogram and mean with the sample histogram and mean obtained in part a. Explain why it is possible for the sample and the population to be dissimilar.

Please note that the actual calculations and analysis need to be performed using the appropriate statistical software/tool or spreadsheet program that supports statistical functions.