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A computer company that recently developed a new software product wanted to estimate the mean time taken to learn how to use this software by people who are somewhat familiar with computers. A random sample of 12 such persons was selected. The following data give the times taken (in hours) by these persons to learn how to use this software. $$ \begin{array}{llllll} 1.75 & 2.25 & 2.40 & 1.90 & 1.50 & 2.75 \\ 2.15 & 2.25 & 1.80 & 2.20 & 3.25 & 2.60 \end{array} $$ Construct a \(95 \%\) confidence interval for the population mean. Assume that the times taken by all persons who are somewhat familiar with computers to learn how to use this software are approximately normally distributed.

Step by step solution

01

Calculate the Sample Mean and Sample Standard Deviation

First, the sample mean (also called the sample average) and the sample standard deviation have to be calculated. The sample mean is the sum of all the data values divided by the number of data values. The sample standard deviation is the square root of the sum of the squared differences between each data value and the sample mean divided by the (number of data values - 1).

02

Find the Z-Score

The z-score for a 95% confidence interval is the value that cuts off the upper 2.5% of the standard normal distribution, since 5% is to be cut off from the total area under the curve, leaving 95% for the central area. The z-score for two-tailed test at 95% confidence level is approximately 1.96.

03

Calculate the Confidence Interval

The formula to calculate a confidence interval is (X̄ - Z * (s/√n), X̄ + Z * (s/√n)), where X̄ is the sample mean, Z is the z-score, s is the sample standard deviation, and n is the number samples. Substitute the previously calculated values into this formula to find the confidence interval.

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Key Concepts

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

Sample Mean Calculation

The sample mean, often represented as \( \bar{X} \), is a central value that gives us an idea of the average time it takes for individuals to learn the software in our given sample.
To calculate the sample mean, we simply sum up all the individual data points and divide by the number of data points.
Mathematically, this is expressed as:

• \( \bar{X} = \frac{1.75 + 2.25 + 2.40 + 1.90 + 1.50 + 2.75 + 2.15 + 2.25 + 1.80 + 2.20 + 3.25 + 2.60}{12} \)

This results in an average time which represents the central tendency of our data.
By calculating the sample mean, we aim to approximate the population mean, which is often unknown in practice.

Sample Standard Deviation

The sample standard deviation provides us an understanding of how spread out the data points are in relation to the sample mean.
It represents the average distance between each data point and the mean. This helps to understand the variability in learning times.
The formula for the sample standard deviation \( s \) is:

• \( s = \sqrt{\frac{\sum (x_i - \bar{X})^2}{n-1}} \)

Where:

• \( x_i \) represents each data point
• \( \bar{X} \) is the sample mean
• \( n \) is the number of data points, which in this case is 12

The calculation involves finding the square of the difference between each data point and the mean, summing those squares, dividing by \( n-1 \), and finally taking the square root.
This concept of measuring spread is crucial, as it affects our confidence interval width.

Z-Score

The z-score is an essential statistical tool that helps us determine how data points relate to the standard normal distribution.
In constructing confidence intervals, we use it to identify the appropriate cut-off points on the normal curve.
For a 95% confidence interval, we need a z-score that includes the middle 95% of the distribution.
This leaves 2.5% in each tail, and the corresponding z-score is approximately 1.96 for a two-tailed test.

• This systematic selection of the z-score makes sure the confidence interval covers the true population mean with 95% certainty.
• The z-score is critical as it adjusts the interval width based on desired confidence levels.

Understanding the statistical significance of the z-score ensures that we can correctly interpret and scale our intervals for real-world applications.

Normal Distribution

Normal distribution, often called the bell curve, is a fundamental concept in statistics describing a symmetric, bell-shaped curve where most of the observations cluster around the central peak.
In this context, we assume our data is approximately normally distributed, which helps in constructing valid confidence intervals.
This assumption allows us to use the z-score method effectively.

• Normal distribution has mean \( \mu \) and standard deviation \( \sigma \).
• The majority of data falls within 68% of one standard deviation from the mean, 95% within two, and 99.7% within three standard deviations, which is the Empirical Rule.

Recognizing the properties of a normal distribution is crucial since it affects statistical procedures, such as hypothesis testing and confidence intervals.
This assumption is especially key when the sample size is small, like in this exercise.