Question

The contents of seven similar containers of sulfuric acid are \(9.8,10.2,10.4\), \(9.8,10.0,10.2\), and \(9.6\) liters. Find a \(95 \%\) confidence interval for the mean of all such containers, assuming an appropriate normal distribution.

Short answer

The 95% confidence interval for the mean of all such sulfuric acid containers is approximately between 9.79 and 10.21 liters.

Step-by-step solution

Compute the Mean of Sample Data

To compute the mean, sum up all the liters of sulfuric acid and divide this sum by the number of containers. This will give the sample average (mean). The containers total are \(9.8, 10.2, 10.4, 9.8, 10.0, 10.2, 9.6\) liters. The mean \(\bar{x}\) is given by \(\frac{70}{7} = 10\) liters.

Calculate the Standard Deviation

To calculate the sample standard deviation (s), subtract the mean from each observation, square the result, sum them, divide by the number of observations minus 1 and take the square root of the result. The standard deviation is \(\sqrt{\frac{{(9.8-10)^2+(10.2-10)^2+(10.4-10)^2+(9.8-10)^2+(10.0-10)^2+(10.2-10)^2+(9.6-10)^2}}{6}} = 0.24\) liters.

Determine the Z-Score for the 95% Confidence Level

A 95% confidence interval corresponds to 1.96 standard deviations away from the mean in a normal distribution. The critical z-value is \(\pm 1.96\).

Calculate the Confidence Interval

The confidence interval for the mean of all such containers is given by mean \(\pm\) (critical z-value) \(\times\) (standard deviation / square root of sample size). This results in \(10 \pm 1.96 \times (0.24 / \sqrt{7})\), yielding an interval of \(9.79, 10.21\) liters.

Key concepts

Sample Mean

The sample mean is a measure that represents the average value of a sample. It provides a central location for the data. To find the sample mean, you add up all the data points and divide by the number of points. In the exercise, the data points are the amounts of sulfuric acid in each container. Summing them gives you a total of 70 liters. Dividing this sum by the number of containers (7) results in a sample mean of 10 liters.

Understanding the sample mean is crucial as it acts as an estimate of the actual mean of the whole population. It's a basic yet powerful way to see where most of your data points are clustered. Always keep in mind that this is just an estimate, and it depends on the sample you have.

Standard Deviation

Standard deviation is a measure that indicates how much the data points deviate from the mean, or on average, how far they are from that central point. To find this measure, you first calculate the difference between each data point and the mean, square that difference, and find the average of those squared differences. Finally, take the square root of that average.

In our solution, we did these calculations with the mean value of 10, yielding a standard deviation of 0.24 liters. A smaller standard deviation implies that the data points are close to the mean, whereas a larger one indicates a wider spread. This concept is important because it shows the variability within your data set, providing insights into the reliability of your sample mean as a representation of the population mean.

Normal Distribution

The normal distribution is a bell-shaped curve that is symmetric around the mean, representing how data is typically spread. Many data sets in the real world tend to gain this shape, which is why the normal distribution is a key concept in statistics.

In the context of this exercise, assuming a normal distribution allows us to use the properties of this distribution to make inferences about the population from our sample. Specifically, it enables us to calculate confidence intervals, like the 95% interval used here. This assumption can sometimes be challenging with small sample sizes, but it generally holds with larger samples or when we have historical data to support it.

Z-Score

A Z-score tells you how many standard deviations a data point is from the mean. This is a universal measure used to compare data points from different distributions. In the context of confidence intervals, the Z-score helps determine how wide the confidence interval needs to be to capture the population mean with a certain level of confidence.

For a 95% confidence interval, the critical Z-score value is approximately 1.96. This means we want our interval to capture the mean 1.96 standard deviations below and above our sample mean. By applying the Z-score, we ensure that this coverage is statistically sound, giving us more context about where the true mean might fall in the wider population.