Question

According to the Vivino website, suppose the mean price for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is $32.48. A New England-based lifestyle magazine wants to determine if red wines of the same quality are less expensive in Providence, and it has collected prices for 60 randomly selected red wines of similar quality from wine stores throughout Providence. The mean and standard deviation for this sample are $30.15 and $12, respectively.

(a)
Develop appropriate hypotheses for a test to determine whether the sample data support the conclusion that the mean price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48. (Enter != for ≠ as needed.)
H0:

Ha:

(b)
Using the sample from the 60 bottles, what is the test statistic? (Round your answer to three decimal places.)

Using the sample from the 60 bottles, what is the p-value? (Round your answer to four decimal places.)
p-value =
(c)
At α = 0.05, what is your conclusion?
Do not reject H0. We can conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.
Reject H0. We can conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.
Do not reject H0. We cannot conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.
Reject H0. We cannot conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.
(d)
Repeat the preceding hypothesis test using the critical value approach.
State the null and alternative hypotheses. (Enter != for ≠ as needed.)
H0:

Ha:

Find the value of the test statistic. (Round your answer to three decimal places.)

State the critical values for the rejection rule. Use
α = 0.05.
(Round your answers to three decimal places. If the test is one-tailed, enter NONE for the unused tail.)
test statistic ≤
test statistic ≥
State your conclusion.
Do not reject H0. We can conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.
Reject H0. We can conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.
Do not reject H0. We cannot conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.
Reject H0. We cannot conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.

Answer 1

The test determines whether the mean price in Providence for red wine is less than the population mean. The null and alternative hypotheses are set, the test statistic and p-value are calculated, and the conclusion is drawn based on the significance level. The hypothesis test is then repeated using the critical value approach.

Step-by-step explanation:

(a) The null and alternative hypotheses for this test are:
H0: The mean price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is equal to or greater than the population mean of $32.48.
Ha: The mean price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.

(b) The test statistic, t, can be calculated using the formula:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
Using the given values, the test statistic is approximately -1.865.

(c) The p-value can be determined by finding the area under the t-distribution curve to the left of the test statistic. In this case, the p-value is approximately 0.0337.

(d) The critical value approach requires comparing the test statistic to the critical values. Since the test is one-tailed and the significance level is 0.05, the critical value is approximately -1.671. Therefore, the rejection region is t ≤ -1.671. Based on the test statistic of -1.865, we reject the null hypothesis.

At α = 0.05, the correct conclusion is to Reject H0. We can conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.