Ch. 2 - Descriptive Statistics

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경

Larson 8th Edition

Ch. 2 - Descriptive Statistics

문제 2.CR.6b

2장, 문제 2.CR.6b

The mean annual salary for a sample of electrical engineers is \$86,500, with a standard deviation of $1500. The data set has a bell-shaped distribution.

b. The salaries of three randomly selected electrical engineers are $93,500, \$85,600, and \$82,750. Find the z-score that corresponds to each salary. Determine whether any of these salaries are unusual.

검증된 단계별 안내

Step 1: Recall the formula for calculating the z-score:

z = (x - μ) / σ

, where

x

is the data point,

μ

is the mean, and

σ

is the standard deviation.

Step 2: Substitute the given values into the formula for each salary. For the first salary, \(93,500, calculate

z = (93500 - 86500) / 1500

. For the second salary, \)85,600, calculate

z = (85600 - 86500) / 1500

. For the third salary, \$82,750, calculate

z = (82750 - 86500) / 1500

Step 3: Simplify the numerator for each calculation. For the first salary, compute

93500 - 86500

. For the second salary, compute

85600 - 86500

. For the third salary, compute

82750 - 86500

Step 4: Divide the result of each numerator by the standard deviation,

1500

, to find the z-scores for each salary.

Step 5: Determine whether any of the z-scores are unusual. Recall that a z-score is considered unusual if it is less than -2 or greater than 2. Compare each calculated z-score to this threshold to identify any unusual salaries.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Z-Score

A Z-score measures how many standard deviations a data point is from the mean of a dataset. It is calculated using the formula Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. A Z-score helps in identifying how unusual or typical a value is within a distribution, particularly in a normal distribution.

Standard Deviation

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 that the values are spread out over a wider range. In this context, the standard deviation of $1500 provides insight into the variability of electrical engineers' salaries around the mean of $86,500.

Normal Distribution

A 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. In this scenario, the mention of a bell-shaped distribution implies that the salaries of electrical engineers follow this pattern, allowing for the application of Z-scores to assess the unusualness of specific salaries.