91Ó°ÊÓ

Daily Activity. It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking. I Among mildly obese people, the mean number of minutes of daily activity (standing or walking) is approximately Normally distributed with mean 373 minutes and standard deviation 67 minutes. The mean number of minutes of daily activity for lean people is approximately Normally distributed with mean 526 minutes and standard deviation 107 minutes. A researcher records the minutes of activity for an SRS of five mildly obese people and an SRS of five lean people. a. What is the probability that the mean number of minutes of daily activity of the five mildly obese people exceeds 420 minutes? b. What is the probability that the mean number of minutes of daily activity of the five lean people exceeds 420 minutes?

Step by step solution

01

Identify parameters for mildly obese people

Given that for mildly obese people, the mean (\( \mu_1 \) ) number of minutes is 373 and the standard deviation (\( \sigma_1 \) ) is 67 minutes, and we have a sample size of 5.

02

Calculate the standard error for mildly obese people

Use the formula for the standard error: \( SE_1 = \frac{\sigma_1}{\sqrt{n}} \). Substituting the values, \( SE_1 = \frac{67}{\sqrt{5}} \approx 29.97 \).

03

Calculate the Z-score for mildly obese sample

To determine the Z-score, use the formula \( Z = \frac{\bar{x} - \mu}{SE} \), where \( \bar{x} \) is 420 minutes. Substituting, we have \( Z = \frac{420 - 373}{29.97} \approx 1.57 \).

04

Find probability for mildly obese people

Using standard normal distribution tables or a calculator, find the probability of getting a Z-score greater than 1.57. This probability corresponds to about 0.0582.

05

Identify parameters for lean people

For lean people, the mean (\( \mu_2 \) ) is 526 minutes and the standard deviation (\( \sigma_2 \) ) is 107 minutes, with a sample size of 5.

06

Calculate the standard error for lean people

Use the formula for the standard error: \( SE_2 = \frac{\sigma_2}{\sqrt{n}} \). Substituting the values, \( SE_2 = \frac{107}{\sqrt{5}} \approx 47.86 \).

07

Calculate the Z-score for lean sample

Using the formula \( Z = \frac{\bar{x} - \mu}{SE} \), where \( \bar{x} \) is 420 minutes, we get \( Z = \frac{420 - 526}{47.86} \approx -2.21 \).

08

Find probability for lean people

Using standard normal distribution tables or a calculator, find the probability of getting a Z-score greater than -2.21. This probability corresponds to about 0.9864.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

The Normal Distribution is a fundamental concept in statistics. It describes how data tends to group around a central value with a symmetrical "bell curve" shape.
It's crucial for understanding variations in daily activities among different groups of people. For instance, when analyzing daily activities of mildly obese and lean people, their minutes of activity are treated as normally distributed data.
The mean is the peak of this bell curve, and most values tend to cluster around it.

• The shape of the curve is symmetrical.
• It's defined by two parameters: the mean and the standard deviation.
• 68% of data falls within one standard deviation, 95% within two, and 99.7% within three standard deviations.

Recognizing data as normally distributed allows for predictions and probability calculations, like determining the likelihood of certain activity levels.

Z-score Calculation

The Z-score is a measure that describes a data point's position relative to the mean of a group of values, expressed in terms of standard deviations.
In practical scenarios like assessing daily activity, it helps determine how extreme an observation is when compared to the average.

Calculation of Z-score

Here's the formula for calculating the Z-score:
\[ Z = \frac{\bar{x} - \mu}{SE} \]- \( \bar{x} \) is the sample mean,- \( \mu \) is the population mean,- \( SE \) is the standard error.A positive Z-score indicates a value above the average, while a negative one shows it's below.
This is especially useful in comparing two groups, like mildly obese and lean individuals, to find the probability of observing certain activity levels.

Standard Error

The Standard Error (SE) quantifies how much the sample mean is expected to vary from the true population mean.
It is especially helpful when we need to make predictions about a population based on a sample. Understanding this helps in accurately assessing daily activity levels in different populations.

Formula for Standard Error

The formula to calculate SE is:\[ SE = \frac{\sigma}{\sqrt{n}} \]- \( \sigma \) is the population standard deviation,- \( n \) is the sample size.The smaller the standard error, the more representative the sample mean is of the population mean.
It decreases as sample size increases, reflecting higher accuracy with larger samples.

Sample Size

Sample Size ( ) refers to the number of observations in a sample. It's essential when generalizing the results of a study to the greater population.
In our context, we explored a sample size of 5 for both mildly obese and lean groups. With such groups, calculating probabilities about daily activities can help understand broader trends.

The Impact of Sample Size

A larger sample size generally increases the reliability of statistical estimates.

• Smaller samples yield more variation and less reliable estimates.
• Larger samples allow for more precise predictions about a population.

However, larger sizes can be more resource-intensive, requiring a balance between precision and practicality.

Standard Deviation

Standard Deviation is a measure of the dispersion or variability within a dataset.
When we discuss daily activity minutes for different groups of people, standard deviation illustrates how much individual activity times vary from the mean.

Concept of Standard Deviation

Standard deviation is denoted as \( \sigma \) and can be calculated using:\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \]- \( x_i \) represents each data point,- \( \mu \) the mean,- \( N \) the total number of data points.Smaller standard deviation means data points are close to the mean, indicating less variability.
Larger values suggest more spread out data, which can be vital in analyzing how spread out daily activity levels are across different groups like mildly obese and lean people.