Seventy efficiency apartments were randomly Seventy efficiency apartments were randomly sampled in a college town. The monthly rent prices for the apartments are listed below in ascending order. Distribution Shape: Skewness n Example: Apartment Rents

The z-score is often called the standardized value. The z-score is often called the standardized value. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. z-Scores Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score. Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score.

z-Scores A data value less than the sample mean will have a A data value less than the sample mean will have a z-score less than zero. z-score less than zero. A data value greater than the sample mean will have A data value greater than the sample mean will have a z-score greater than zero. a z-score greater than zero. A data value equal to the sample mean will have a A data value equal to the sample mean will have a z-score of zero. z-score of zero. An observation’s z-score is a measure of the relative An observation’s z-score is a measure of the relative location of the observation in a data set. location of the observation in a data set.

Chebyshev’s Theorem At least (1 - 1/ z 2 ) of the items in any data set will be At least (1 - 1/ z 2 ) of the items in any data set will be within z standard deviations of the mean, where z is within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1. At least (1 - 1/ z 2 ) of the items in any data set will be At least (1 - 1/ z 2 ) of the items in any data set will be within z standard deviations of the mean, where z is within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1. Chebyshev’s theorem requires z > 1, but z need not Chebyshev’s theorem requires z > 1, but z need not be an integer. be an integer. Chebyshev’s theorem requires z > 1, but z need not Chebyshev’s theorem requires z > 1, but z need not be an integer. be an integer.

At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 75%75% z = 2 standard deviations z = 2 standard deviations Chebyshev’s Theorem At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 89%89% z = 3 standard deviations z = 3 standard deviations At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 94%94% z = 4 standard deviations z = 4 standard deviations

Empirical Rule When the data are believed to approximate a When the data are believed to approximate a bell-shaped distribution … bell-shaped distribution … The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean.

Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean %68.26% +/- 1 standard deviation of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean %95.44% +/- 2 standard deviations of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean %99.72% +/- 3 standard deviations

Empirical Rule x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%

Detecting Outliers An outlier is an unusually small or unusually large An outlier is an unusually small or unusually large value in a data set. value in a data set. A data value with a z-score less than -3 or greater A data value with a z-score less than -3 or greater than +3 might be considered an outlier. than +3 might be considered an outlier. It might be: It might be: an incorrectly recorded data value an incorrectly recorded data value a data value that was incorrectly included in the a data value that was incorrectly included in the data set data set a correctly recorded data value that belongs in a correctly recorded data value that belongs in the data set the data set

A survey of local companies found that the mean amount of travel allowances for executives was 0.25 USD per mile. The standard deviation was 0.02 USD. Using Chebyshev`s Theorem find the minimum percentage of the data values that will fall between 0.20 USD and 0.30 USD.

Problem Based on a survey of dental practitioners, the study reported that the mean number of units of local anesthetics used per week by dentists was 79, with a standard deviation of 23. Suppose we want to determine the percentage of dentists who use less than 102 units of local anesthetics per week. a- Assuming nothing is known about the shape of the distribution for the data, what percentage of dentists use less than 102 units of local anesthetics per week? b- Assuming that the data has a mound-shaped (bell-shaped or symmetric) distribution, what percentage of dentists use less than 102 units of local anesthetics per week?

Problem Based on the study to compare the effectiveness of washing the hands with soap and rubbing the hands with alcohol-based antiseptics. Table: Descriptive statistics on bacteria counts for the two groups of health care workers. MeanStandard Deviation Hand rubbing3559 Hand washing69106 a- For hand rubbers, form an interval that contains at least 75% of the bacterial counts. b- For hand washers, form an interval that contains at least 75% of the bacterial counts. (Note that the bacterial count cannot be less than 0) c- On the basis of your results in parts a and b, make an inference about the effectiveness of the two hand cleaning methods.