To solve this problem, let’s work through each part of the question step-by-step. Here are the given relationships for ratings by experts and nonexperts:

• For experts: $y = -0.048 \ln(x) + 2.337$
• For nonexperts: $y = 0.09 \ln(x) + 1.887$

where:

• $y$ is the rating of the wine,
• $x$ is the price of the wine in dollars.

Let’s tackle each part:

(a) What is the rating of wine priced at $50 per bottle for the experts?

For experts: $y = -0.048 \ln(50) + 2.337$

1. Calculate $\ln(50)$.
2. Substitute this value into the formula and solve for $y$.

(b) What is the rating of wine priced at $50 per bottle for the nonexperts?

For nonexperts: $y = 0.09 \ln(50) + 1.887$

1. Use the $\ln(50)$ value from part (a).
2. Substitute into the formula and solve for $y$.

(c) What is the price of a wine rated 2.4 for the experts?

For experts, set $y = 2.4$: $2.4 = -0.048 \ln(x) + 2.337$

1. Isolate $\ln(x)$ and solve for $x$ by using exponentiation.

(d) What is the price of a wine rated 2.4 for the nonexperts?

For nonexperts, set $y = 2.4$: $2.4 = 0.09 \ln(x) + 1.887$

1. Isolate $\ln(x)$ and solve for $x$ by using exponentiation.

(e) At what price are the ratings of the two wines equal? What is the rating of the wine at this price?

Set the two equations equal to each other: $-0.048 \ln(x) + 2.337 = 0.09 \ln(x) + 1.887$

1. Combine terms involving $\ln(x)$.
2. Solve for $x$, then substitute $x$ back into either equation to find the rating.

(f) Explain the effect of the coefficients for each of the two (expert and nonexpert) models.

The coefficients of $\ln(x)$ indicate how sensitive each group's rating is to changes in price. The positive coefficient for nonexperts (0.09) suggests their rating increases as price increases, while the negative coefficient for experts (-0.048) suggests a decrease, indicating differing perceptions of quality with price.

Would you like the detailed calculations for each part?

Here are five related questions to consider:

1. How would changing the price to a different amount, like $100, affect the ratings for experts and nonexperts?
2. What does a positive coefficient of $\ln(x)$ imply about perception of quality with price?
3. How would the model change if experts had a positive coefficient as well?
4. Can these logarithmic models predict ratings accurately for very low or very high prices?
5. What does the intercept term (like 2.337 for experts) represent in these equations?

Tip: Logarithmic functions are useful for modeling relationships where growth rate changes at different scales, like diminishing returns in perceived value.