Question

According to the Vivino website, 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 Englandâbased lifestyle magazine wants to determine if red wines of the same quality are less expensive in Providence, and it has collected prices for 56 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.

Required:
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.
b. Using the sample from the 60 bottles, what is the test statistic?
c. At α = 0.05, what is your conclusion?

Answer 1

a

The null hypothesis is
`H_o  :  \mu =  32.48`

The alternative hypothesis is
`H_a  :  \mu  <  32.48`

b

`t  =  -1.504`

c

There is no sufficient evidence to conclude 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

Explanation:

From the question we are told that

The mean price of is
`\mu = \$ 32.48`

The sample size is n = 56

The sample mean is
`\=  x  =  \$30.15`

The standard deviation is
`\sigma =  \$12`

The null hypothesis is
`H_o  :  \mu =  32.48`

The alternative hypothesis is
`H_a  :  \mu  <  32.48`

Generally the test statistic is mathematically represented as

`t =  (\= x  - \mu )/((\sigma )/(√(n) ) )`

=>
`t =  (30.15  - 32.48 )/((12 )/(√(60) ) )`

=>
`t  =  -1.504`

Generally the p-value is mathematically represented as

`p-value  =  P (t < -1.504 )`

From the z-table

`P(t <  -1.504) =   0.066291`

So

`p-value  = 0.066291`

From the value we obtain we see that

`p -value  >  \alpha`

Then we fail to reject the null hypothesis

Hence there is no sufficient evidence to conclude 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