In Exercises 33 and 34, find the indicated probabilities. If convenient, use technology to find the probabilities.


The daily surface concentration of carbonyl sulfide on the Indian Ocean is normally distributed, with a mean of 9.1 picomoles per liter and a standard deviation of 3.5 picomoles per liter. Find the probability that on a randomly selected day, the surface concentration of carbonyl sulfide on the Indian Ocean is


a. between 5.1 and 15.7 picomoles per liter.

Verified step by step guidance

1. Step 1: Identify the key parameters of the normal distribution. The mean (μ) is 9.1 picomoles per liter, and the standard deviation (σ) is 3.5 picomoles per liter. The problem asks for the probability that the concentration is between 5.1 and 15.7 picomoles per liter.
2. Step 2: Standardize the values 5.1 and 15.7 to z-scores using the z-score formula: z = (x - μ) / σ. For 5.1, calculate z₁ = (5.1 - 9.1) / 3.5. For 15.7, calculate z₂ = (15.7 - 9.1) / 3.5.
3. Step 3: Use a standard normal distribution table or technology to find the cumulative probabilities corresponding to z₁ and z₂. Let P₁ be the cumulative probability for z₁ and P₂ be the cumulative probability for z₂.
4. Step 4: Subtract the cumulative probability for z₁ (P₁) from the cumulative probability for z₂ (P₂) to find the probability that the concentration is between 5.1 and 15.7. The formula is P = P₂ - P₁.
5. Step 5: Interpret the result. The value obtained represents the probability that the surface concentration of carbonyl sulfide on a randomly selected day falls within the specified range.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by two parameters: the mean (average) and the standard deviation (which measures the spread of the data). In this context, the surface concentration of carbonyl sulfide follows a normal distribution, allowing us to use statistical methods to find probabilities within specified ranges.

Mean and Standard Deviation

The mean is the average value of a dataset, calculated by summing all values and dividing by the number of values. The standard deviation quantifies the amount of variation or dispersion in a set of values; a low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. In this problem, the mean concentration of carbonyl sulfide is 9.1 picomoles per liter, and the standard deviation is 3.5 picomoles per liter, which are essential for calculating probabilities.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined set of outcomes. For normally distributed data, this often requires converting raw scores into z-scores, which represent the number of standard deviations a value is from the mean. By using z-scores and standard normal distribution tables or technology, we can find the probability that the surface concentration of carbonyl sulfide falls within a specified range, such as between 5.1 and 15.7 picomoles per liter.