Question

A debate rages whether or not bottled water is worth the cost. The National Resource Defense Council concludes, “there is no assurance that bottled water it is any cleaner or safer than tap (water).” But, in addition to the safety issue, many people prefer the taste of bottled water to tap water, or so they say!

Let’s suppose that the city council members of Corvallis are trying to show that the city of Corvallis tap water tastes just as good as bottled water. To attempt to show this, the city council randomly samples 250 Corvallis, Oregon residents. Each resident participates in a taste test where they are brought individually into a room that contains two cups of water, one of which is bottled water and the other is city of Corvallis tap water. The participant does not know which cup contains the tap or bottled water. The participant tastes the water in each cup and then reports to the administrator the cup that contained the water they thought tasted better.

Suppose 135 of the 250 participants preferred the taste of bottled water over city of Corvallis tap water. Is this evidence that Corvallis residents prefer the taste of bottled water? To answer this question of interest, answer the following questions.

1.   What is the random variable in this problem? Does the random variable have a binomial distribution? Explain. (Recall, there are 4 checks for a discrete random variable to have a binomial distribution – make sure you list and briefly discuss all 4.) (3 points)

2.   What does p represent in the context of this study? (1 point)

3.   Calculate the sample proportion, , of Corvallis residents in the study who prefer the taste of bottled water over city of Corvallis tap water. (1 point)

4.   Perform the hypothesis test in R:

      a.   State the null and alternative hypotheses in statistical notation AND in words!! (3 points)

      b.   Assume that the observations are independent of each other. Which hypothesis test is the appropriate one to use in this situation? Why? (This is where you’d check the condition that the “sample size is large enough”) (2 points)

      c.   Using R, perform the appropriate hypothesis test. Report the z-statistic and p-value (but do not include the R output!). (Note: if the binomial formula was used, no test-statistic needs to be given.) (1 point)

      d.   Based on the p-value, answer the question of interest in a complete sentence in the context of the problem. (3 points)

5.   Suppose you wanted to estimate the proportion of all Corvallis residents who prefer the taste of bottled water over city of Corvallis tap water. If an estimate of the population proportion is desired, a confidence interval should be constructed. Use R to construct a 90% confidence for the proportion of all Corvallis residents who prefer the taste of bottled water over city of Corvallis tap water. Report and interpret the confidence interval. (3 points)

6.   a.   By hand, calculate the standard deviation of the sample proportion ( ) used in the hypothesis test. Show work. (2 points)

        b.   By hand, calculate the standard deviation of the sample proportion used in the construction of the confidence interval. Show work. (2 points) (Technical note: this is actually called a standard error.)

      c.   Why does the formula for the standard error when doing a confidence interval use the sample proportion instead of the hypothesized value of the population proportion? (1 point)

7.   The one-sample z-test for proportions determines the p-value using the “normal approximation method”. This method approximates the exact p-value that would have been found if the binomial formula were used. But, the “normal approximation method” only works well when the sample size is “large enough”.

      a.   If the hypothesis test is done by hand, why do you think we use the “normal approximation method” for large sample sizes instead of using the binomial formula to find p-values? (2 points)

      b    Why do you think it’s not as necessary today to use the normal approximation method compared to, say, 50 years ago? (2 points)

Answer 1

solving first 4

1

the proprotion of people who prefer bottled water is the RV here

the varaible has a binomial distribution because there are only two possible outcomes and all trials are randomly sampled

2.

2

p refers to the proportion of people who prefer bolied water

3

p =135/250 = .54

4

ho = p bottled = p tap

same proportion of people prefer bottled and tap

ha p bottled > p tap

more people prefer bottled to tap

we can do a binomial test

as sample size is large here we can also consider a z test.