Question

how much soda?
the smithville bottling company claims that each can of soda it produces contains, on average, 8 ounces of liquid. however, because the factorys machinery is old, there is some variation. some cans may have a little more than 8 ounces, while others may have a little less. on average, though, there are 8 ounces of soda in a can with a standard deviation of 0.2 ounces.
1: how would you design an experiment to determine if there really are, on average, 8 ounces of soda in each can?

Explanation:

1. Random Sampling: Randomly select a sufficient number of soda - cans from the production line. This helps to ensure that the sample is representative of the entire population of cans.
2. Measurement: Use a precise measuring device to measure the amount of liquid in each selected can.
3. Calculation of Sample Statistics: Calculate the sample mean and sample standard deviation of the measured amounts of liquid in the cans.
4. Hypothesis Testing: Set up a null hypothesis \(H_0:\mu = 8\) (where \(\mu\) is the population mean amount of liquid in a can) and an appropriate alternative hypothesis (e.g., \(H_1:\mu

eq8\)). Then, use a t - test or z - test depending on the sample size and whether the population standard deviation is known (here we know the population standard deviation \(\sigma = 0.2\)) to determine if the sample data provides enough evidence to reject the null hypothesis.

Answer:

1. Randomly select a large number (e.g., \(n = 100\)) of soda - cans from different production batches.
2. Measure the amount of liquid in each of the selected cans using a calibrated measuring cup or similar precise device.
3. Calculate the sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\), where \(x_i\) is the amount of liquid in the \(i\) - th can and \(n\) is the sample size. Also calculate the sample standard deviation \(s\) if not using the known population standard deviation \(\sigma = 0.2\).
4. Conduct a hypothesis test. If the sample size \(n\) is large (\(n\geq30\)) and the population standard deviation \(\sigma = 0.2\) is known, use the z - test statistic \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\mu = 8\). If the sample size is small (\(n\lt30\)) and the population standard deviation is unknown, use the t - test statistic \(t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\). Compare the calculated test statistic with the critical value from the appropriate distribution (z - distribution or t - distribution) at a chosen significance level (e.g., \(\alpha = 0.05\)) to determine if there is evidence to reject the claim that the average amount of soda in each can is 8 ounces.