The weight of a sophisticated running shoe is normally...

The weight of a sophisticated running shoe is normally distributed with a mean of 12 ounces and a standard deviation of 0.5 ounce. (a) What is the probability that a shoe weighs more than 13 ounces? (b) What must the standard deviation of weight be in order for the company to state that $99.9 \%$ of its shoes are less than 13 ounces? (c) If the standard deviation remains at 0.5 ounce, what must the mean weight be in order for the company to state that $99.9 \%$ of its shoes are less than 13 ounces?

The weight of a sophisticated running shoe is normally distributed with a mean of 12 ounces and a standard deviation of 0.5 ounce.
(a) What is the probability that a shoe weighs more than 13 ounces?
(b) What must the standard deviation of weight be in order for the company to state that $99.9 \%$ of its shoes are less than 13 ounces?
(c) If the standard deviation remains at 0.5 ounce, what must the mean weight be in order for the company to state that $99.9 \%$ of its shoes are less than 13 ounces?

Key Concepts

Normal Distribution

The normal distribution is a symmetric, bell-shaped probability distribution characterized by its mean and standard deviation. It is fundamental in statistics due to the Central Limit Theorem, which states that the distribution of sample means tends to be normal regardless of the distribution of the original data under certain conditions.

Z-Score Transformation

The z-score is a standardized measure that indicates how many standard deviations a particular value is from the mean of the distribution. Calculating the z-score allows us to transform a normally distributed variable into the standard normal distribution, which facilitates finding probabilities and percentiles.

Probability Computation in Normal Distribution

This concept involves calculating the likelihood that a normally distributed random variable falls above, below, or between specified values. It typically requires converting the values into z-scores and using the cumulative distribution function for the standard normal distribution to obtain probabilities.

Percentiles and Inverse Cumulative Distribution Function

Percentiles indicate the relative standing of a value within a distribution, describing the percentage of data that falls below a certain point. The inverse cumulative distribution function (or quantile function) is used to determine the value corresponding to a given percentile, which is crucial in setting thresholds such as ensuring a specified percentage of observations lies below a certain limit.

Standard Deviation Adjustment

Modifying the standard deviation alters the spread of the normal distribution. A smaller standard deviation means data points are clustered closely around the mean, while a larger standard deviation indicates a wider spread. Adjusting the standard deviation is essential when modifying the distribution to ensure a particular percentage of observations falls below or above a specific value.

Mean Adjustment

Altering the mean shifts the entire normal distribution left or right without changing its spread. This adjustment is used to reposition the distribution relative to a threshold value, such as ensuring that a specific percentage of the data falls below a certain weight or measurement.

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