Understanding Normal Distribution in Price Comparison: A Statistical Analysis

The concept of normal distribution is widely used in various fields, including finance, economics, marketing, and sales. This article will explore how normal distribution can be applied in price comparison between televisions. We will use statistical methods to understand the pricing patterns and probabilities of price differences between two brands of televisions.

NORMAL DISTRIBUTION AND TELEVISION PRICES

Suppose that prices of a brand of television are normally distributed with a mean of 400 and a standard deviation of 60. The normal distribution, often referred to as the bell curve, is a continuous probability distribution that is symmetric around the mean (μ). In this case, μ 400 and σ 60.

Let's delve into the problem: If a particular television is cheaper than 90 of all other televisions of this brand, how much does it cost?

Cheap Television Cost

Since the prices are normally distributed, 10% of the televisions would be cheaper. To find the price at the 10th percentile, we can use the z-score formula:

[ z frac{x - mu}{sigma} ]

From the standard normal distribution table, the z-score corresponding to the 10th percentile is approximately -1.28. Thus, we can calculate the price (x) at the 10th percentile:

[ -1.28 frac{x - 400}{60} ]

Solving for x:

[ x 400 - 1.28 times 60 ] [ x 400 - 76.8 ] [ x approx 323.2 ]

Therefore, a television cost ( approx 323.2 ) dollars to be cheaper than 90% of the televisions of this brand.

ANALYSIS OF ANOTHER BRAND OF TELEVISIONS

Consider a different brand of television with a mean price of 300 and a standard deviation of 40. Following the same approach, let's find the probability that the sum of prices for an independently bought television from each brand is less than 600.

Sum of Prices

When two independent normal random variables are added, the resulting sum is also normally distributed. Let's denote the prices of the first brand as ( X ) and the second brand as ( Y ). Then, ( X sim N(400, 60^2) ) and ( Y sim N(300, 40^2) ).

The sum ( Z X Y ) is normally distributed with:

[ mu_Z mu_X mu_Y 400 300 700 ] [ sigma_Z^2 sigma_X^2 sigma_Y^2 60^2 40^2 3600 1600 5200 ] [ sigma_Z sqrt{5200} approx 72.11 ]

Now, we need to find the probability that ( Z [ Z sim N(700, 72.11^2) ]

Using the z-score formula:

[ z frac{600 - 700}{72.11} approx -1.386 ]

The probability corresponding to this z-score is approximately 0.0834 (from the standard normal distribution table).

Conclusion

Based on the analysis, we have found that a television from the first brand is priced approximately at 323.2 dollars to be cheaper than 90% of the televisions of this brand. Furthermore, the probability that the sum of the prices of a television from each brand is less than 600 dollars is about 8.34%.

Understanding price distributions and probabilities can help in making informed decisions and predictions in various business contexts. By leveraging the principles of normal distribution, businesses can optimize pricing strategies and better understand market dynamics.