Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

A new type of car tire is launched by a tire manufacturer. The automobile association performs a durability test on a random sample of 18 of these tires. For each tire the durability is expressed as a percentage: a score of \(100(\%)\) means that the tire lasted exactly as long as the average standard tire, an accepted comparison standard. From the multitude of factors that influence the durability of individual tires the assumption is warranted that the durability of an arbitrary tire follows an \(N\left(\mu, \sigma^{2}\right)\) distribution. The parameters \(\mu\) and \(\sigma^{2}\) characterize the tire type, and \(\mu\) could be called the durability index for this type of tire. The automobile association found for the tested tires: \(\bar{x}_{18}=195.3\) and \(s_{18}=16.7\). Construct a \(95 \%\) confidence interval for \(\mu\).

Step-by-Step Solution

Answer

The 95% confidence interval for \( \mu \) is (187.0, 203.6).

Step 1: Identify the Statistical Properties

We have a sample of 18 tires, so the sample size is \( n = 18 \). The sample mean is given as \( \bar{x}_{18} = 195.3 \), and the sample standard deviation is \( s_{18} = 16.7 \). We need these values to construct the confidence interval for the population mean \( \mu \).

Step 2: Determine the Critical Value

Since we are dealing with a sample size of \( n = 18 \) which is less than 30, we use the t-distribution. For a 95% confidence interval and \( n - 1 = 17 \) degrees of freedom, we look up the critical value (\( t_{\alpha/2} \)) in the t-table or use a calculator. This gives us \( t_{0.025, 17} \approx 2.110 \).

Step 3: Calculate the Standard Error

The standard error (SE) of the sample mean is calculated using the formula: \[SE = \frac{s}{\sqrt{n}} = \frac{16.7}{\sqrt{18}} \approx 3.937\]This represents the standard deviation of the sample mean.

Step 4: Construct the Confidence Interval

The confidence interval for \( \mu \) is given by:\[\bar{x} \pm t_{\alpha/2} \times SE\]Plugging in the values, we get:\[195.3 \pm 2.110 \times 3.937\]Calculating this, the interval is:\[195.3 \pm 8.3037 = (187.0, 203.6)\]Thus, the 95% confidence interval for \( \mu \) is (187.0, 203.6).

Step 5: Interpret the Confidence Interval

Based on the results, we can say with 95% confidence that the true mean durability index (\( \mu \)) for the new type of tire is between 187.0% and 203.6% compared to the standard tire.

Key Concepts

t-distribution

When constructing a confidence interval for a population mean with a small sample size, typically when the sample size is less than 30, the t-distribution is used. This distribution is similar to the normal distribution but has heavier tails, which accounts for the increased variability inherent with small sample sizes.
The degree of freedom plays a crucial role in shaping the t-distribution, with fewer degrees of freedom resulting in even heavier tails. This makes the t-distribution more conservative than the normal distribution for smaller samples.
The critical values from the t-distribution, often denoted as \( t_{\alpha/2} \), are used to calculate the margin of error in the confidence interval. These values are adjusted based on the desired confidence level and the degrees of freedom, which in turn, corresponds to the sample size minus one.

sample mean

The sample mean, denoted as \( \bar{x} \), is a measure of central tendency that provides an estimate of the average value of a population from a sample. In our tire durability example, the sample mean is 195.3, which suggests that, on average, the tires lasted substantially longer than the standard tires.
The sample mean is calculated by adding up all the observed values and dividing by the number of observations. In statistical analysis, the sample mean is used as an unbiased point estimate of the population mean (\( \mu \)). This means that, over many samples, the average of the sample means will equal the population mean.

standard error

The standard error (SE) of the sample mean measures the variability of the sample mean from the actual population mean. Essentially, it indicates how much the sample mean \( \bar{x} \) is expected to vary from sample to sample.
The formula for standard error is \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size. In this context, the standard error provides insight into the precision of the sample mean as an estimate of the population mean.
A smaller standard error indicates that the sample mean is likely to be close to the population mean, increasing confidence in the findings. Conversely, a larger standard error suggests more variability and less precision.

statistical properties

Statistical properties are systematic mathematical characteristics that describe data, often used to make meaningful inferences about a population. Key statistics include mean, variance, and standard deviation, which help summarize data and draw conclusions.
For inferential statistics—such as forming a confidence interval—these properties help us understand the distribution of the data and construct estimates for parameters of the entire population. In constructing the confidence interval for the tire durability study, the statistical properties guide how we estimate the population mean \( \mu \) with a known confidence level.

• This involves assuming a normal distribution of the population data.
• Calculating the sample mean and standard deviation to understand data spread.
• Using statistical properties to properly apply the t-distribution for confidence interval estimation.

Understanding these statistical properties helps ensure accurate and reliable estimations, facilitating better decision-making based on the data.