91Ó°ÊÓ

Data on pull-off force (pounds) for connectors used in an automobile engine application are as follows: 79.3,75.1 , 78.2,74.1,73.9,75.0,77.6,77.3,73.8,74.6,75.5,74.0,74.7 75.9,72.9,73.8,74.2,78.1,75.4,76.3,75.3,76.2,74.9,78.0 75.1,76.8 (a) Calculate a point estimate of the mean pull-off force of all connectors in the population. State which estimator you used and why, (b) Calculate a point estimate of the pull-off force value that separates the weakest \(50 \%\) of the connectors in the population from the strongest \(50 \%\). (c) Calculate point estimates of the population variance and the population standard deviation. (d) Calculate the standard error of the point estimate found in part (a). Interpret the standard error. (e) Calculate a point estimate of the proportion of all connectors in the population whose pull-off force is less than 73 pounds.

Step by step solution

01

Calculate the Mean

To find a point estimate of the mean pull-off force, we calculate the sample mean. The sample mean is the sum of all observations divided by the number of observations. Here, the observations are given as a list of pull-off forces.\[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\]In our dataset, there are 26 values: 79.3, 75.1, 78.2, 74.1, 73.9, 75.0, 77.6, 77.3, 73.8, 74.6, 75.5, 74.0, 74.7, 75.9, 72.9, 73.8, 74.2, 78.1, 75.4, 76.3, 75.3, 76.2, 74.9, 78.0, 75.1, 76.8. Adding these gives 1929.9.Thus, the mean \( \bar{x} = \frac{1929.9}{26} \approx 74.23 \).

02

Find the Median

The point estimate that separates the weakest 50% of the connectors from the strongest is the median. To find the median, we first order the data set from smallest to largest and then find the middle value. Sorting the data gives: 72.9, 73.8, 73.8, 73.9, 74.0, 74.1, 74.2, 74.6, 74.7, 74.9, 75.0, 75.1, 75.1, 75.3, 75.4, 75.5, 75.9, 76.2, 76.3, 76.8, 77.3, 77.6, 78.0, 78.1, 78.2, 79.3. With 26 values, the median is the average of the 13th and 14th values: (75.1 + 75.3) / 2 = 75.2.

03

Calculate Variance and Standard Deviation

Population variance is estimated with the sample variance, calculated as:\[s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}\]First, we calculate each deviation from the mean, square them, sum these squares, and divide by \(n-1\).After performing these calculations, the variance \( s^2 \approx 3.20 \). The standard deviation is the square root of variance: \[ s = \sqrt{s^2} \approx \sqrt{3.20} \approx 1.79 \]

04

Calculate the Standard Error

The standard error of the mean estimates the variability of the sample mean as an estimate of the population mean, calculated by:\[SE = \frac{s}{\sqrt{n}}\]Using our standard deviation \( s \approx 1.79 \) and sample size \( n = 26 \),\[ SE = \frac{1.79}{\sqrt{26}} \approx 0.35 \]. This means the mean's estimate's typical error or variability is about 0.35 pounds.

05

Calculate the Proportion Less Than 73

To find the proportion of all connectors with pull-off force less than 73 pounds, count the number of observations below 73, then divide by the total number of observations. From the sorted data list, the only value less than 73 is 72.9.Thus, the proportion is \( \frac{1}{26} \approx 0.038 \).

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Key Concepts

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

Point Estimate

A point estimate is a single value used to approximate a population parameter. It's like making a best guess based on the data available. In the context of statistics, we often talk about point estimates of the mean, median, variance, or other population characteristics. By focusing on point estimates, statisticians can make informed decisions about the broader population.

However, point estimates do not provide information about the variability or uncertainty surrounding the estimate. Hence, they are often supplemented with confidence intervals or other measures to give a more comprehensive view.

Sample Mean

The sample mean, denoted as \( \bar{x} \), is a point estimate of the population mean. To calculate it, you sum all observations and divide by the number of observations. This formula is represented as:

• \( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \)

In our example, the sample mean is computed based on the 26 observations of pull-off forces. When using the sample mean to estimate the population mean, it's important to remember it may not be perfect, but it offers the most unbiased estimate from available data.

The sample mean helps in understanding the central tendency of the data, and it's a foundational concept in inferential statistics.

Population Variance

Population variance measures how data points differ from the mean. When estimating population variance from a sample, we calculate the sample variance \( s^2 \), utilizing:

• \( s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} \)

Here, each data point's deviation from the mean is squared, summed, and divided by one less than the number of observations \( (n-1) \).

This adjustment, known as Bessel's correction, corrects the bias in the estimation of a population parameter from a sample.

Standard deviation is the square root of the variance, representing the data's spread in the same unit as the original data. Understanding variance and standard deviation is crucial as they provide insights into the data's variability, guiding how much we can rely on the estimate of the mean.

Standard Error

The standard error (SE) indicates how much the sample mean \( \bar{x} \) is expected to vary from the true population mean. It's an essential concept in understanding the reliability of an estimate from sample data. Standard error is derived using the formula:

• \( SE = \frac{s}{\sqrt{n}} \)

where \( s \) is the sample standard deviation and \( n \) is the sample size.

A small standard error suggests that the sample mean is a closer approximation of the population mean.

Conversely, a large standard error implies greater variability, suggesting the sample mean may be less representative.This concept reflects how statistical inference relies on sample-based estimates to make claims about a larger population.