Normal Distribution Using a larger data set than the one given for the preceding exercises, assume that cell phone radiation amounts are normally distributed with a mean of 1.17 W/kg and a standard deviation of 0.29 W/kg.
a. Find the probability that a randomly selected cell phone has a radiation amount that exceeds the U.S. standard of 1.6 W/kg or less.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability that a randomly selected cell phone has a radiation amount exceeding 1.6 W/kg or less, given that the data follows a normal distribution with a mean (μ) of 1.17 W/kg and a standard deviation (σ) of 0.29 W/kg.
2. Step 2: Standardize the value of 1.6 W/kg using the z-score formula: z = (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation. Substitute the given values into the formula.
3. Step 3: Once the z-score is calculated, use the standard normal distribution table (or a statistical software/tool) to find the cumulative probability corresponding to this z-score. This cumulative probability represents the probability that a cell phone has a radiation amount less than or equal to 1.6 W/kg.
4. Step 4: To find the probability that a cell phone has a radiation amount exceeding 1.6 W/kg, subtract the cumulative probability obtained in Step 3 from 1. This is because the total probability for a normal distribution is 1.
5. Step 5: Interpret the result. The final probability represents the likelihood that a randomly selected cell phone has a radiation amount exceeding the U.S. standard of 1.6 W/kg.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing 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 its mean and standard deviation. In this context, the cell phone radiation amounts follow a normal distribution with a specified mean and standard deviation.

Mean and Standard Deviation

The mean is the average of a set of values, representing the central point of a data set. The standard deviation measures the amount of variation or dispersion from the mean. In this question, the mean radiation amount is 1.17 W/kg, and the standard deviation of 0.29 W/kg indicates how much individual radiation amounts typically deviate from this average.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined set of outcomes. In this case, we need to calculate the probability that a randomly selected cell phone has a radiation amount exceeding 1.6 W/kg. This is typically done using the z-score formula, which standardizes the value in relation to the mean and standard deviation, allowing us to use standard normal distribution tables.