Use the following data. The lifetimes of a certain type of...

Use the following data. The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of $100,000 \mathrm{km}$ and a standard deviation of $10,000 \mathrm{km}$ Answer the following questions. In a sample of 5000 of these tires, how many can be expected to last more than $118,000 \mathrm{km} ?$

Use the following data. The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of $100,000 \mathrm{km}$ and a standard deviation of $10,000 \mathrm{km}$
Answer the following questions.
In a sample of 5000 of these tires, how many can be expected to last more than $118,000 \mathrm{km} ?$

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, showing that data near the average are more frequent in occurrence than data far from the average. It is characterized by its bell-shaped curve and completely described by its mean and standard deviation.

Z-Score Transformation

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the individual data point and dividing the result by the standard deviation. This transformation allows the comparison of data points from different normal distributions by converting them to a standard normal distribution.

Tail Probability Calculation

Tail probability calculation involves determining the probability that a random variable from a given distribution falls in the extreme end (or tail) of the distribution. In the context of a normal distribution, this is done by using the Z-score to find the probability from standard normal tables or software, which can then be used to assess how rare or common an event is.

Expected Count Calculation

Expected count calculation uses the probability of a single event occurring to predict the number of occurrences in a larger sample. By multiplying the tail probability (from the normal distribution) by the total sample size, one can estimate the number of instances expected to have a specific property or fall within a specific range.

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