Question

If 36 measurements of the specific gravity of aluminum had a mean of 2.705 and a standard deviation of .028 , find the point estimate for the actual specific gravity of aluminum and calculate the margin of error.

Short answer

Answer: The point estimate for the actual specific gravity of aluminum is 2.705, and the margin of error is approximately 0.00915.

Step-by-step solution

Identify the given values

From the given exercise, we have the following: - Sample size (n) = 36 - Sample mean (x̄) = 2.705 - Sample standard deviation (s) = 0.028

Calculate the point estimate

Since the point estimate is the same as the sample mean, we already have it: Point estimate = x̄ = 2.705

Find the critical z-value

To find the critical z-value, we need to use a standard normal distribution table. Since we want to be 95% confident, we can use the z-value corresponding to 97.5%, which is the value halfway between 95% and 100%. Using a standard normal distribution table or calculator, we find that: Critical z-value = 1.96

Calculate the margin of error

Now, we can use the critical z-value, sample standard deviation, and sample size to calculate the margin of error using the following formula: Margin of error = Critical z-value * (s / √n) Margin of error = 1.96 * (0.028 / √36) Margin of error = 1.96 * (0.028 / 6) Margin of error = 1.96 * 0.00467 Margin of error ≈ 0.00915 So, the point estimate for the actual specific gravity of aluminum is 2.705, and the margin of error is approximately 0.00915.

Key concepts

Specific Gravity of Aluminum

Understanding the concept of specific gravity is crucial when working with different materials. Specific gravity, often symbolized as SG, is a dimensionless quantity that defines the ratio of the density of a substance to the density of a reference substance, typically water at 4 °C, where water has a density of 1 g/cm³. For metals like aluminum, the SG tells us how heavy the metal is compared to water.

For aluminum, which is known for being lightweight yet strong, the specific gravity is an important factor in industries such as aerospace and construction. If an object made of aluminum is submerged in water, the SG can predict whether it will sink or float. A lower SG indicates that the metal is lighter than water. The exercise you provided gives us a mean specific gravity of 2.705 from 36 measurements. This means that aluminum is approximately 2.705 times as dense as water, showing that despite its lightweight reputation, it will sink in water due to its higher density.

Sample Standard Deviation

The sample standard deviation (s) is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In the context of our aluminum measurements, a sample standard deviation of 0.028 signifies that the individual measurements of the specific gravity of aluminum are fairly close to the sample mean of 2.705. This informs us that the sample is consistent with relatively small deviations from the mean, indicating precision in the measurements. It's crucial for students to understand how to calculate and interpret the standard deviation as it serves as the foundation for many other statistical analyses, including the establishment of the margin of error in our given problem.

Critical Z-Value

The critical z-value plays an essential role in hypothesis testing and constructing confidence intervals. It represents the number of standard deviations a data point must be from the mean in order to be considered unusual or surprising in a standard normal distribution.

For a commonly desired confidence level of 95%, the critical z-value helps determine the margin of error for the confidence interval. The value of 1.96, which we get from statistical tables or tools, corresponds to the required z-score that encapsulates the central 95% of the normal distribution. It indicates that we are 95% confident that the true mean (in this case, the actual specific gravity of aluminum) lies within 1.96 standard deviations of our sample mean. This is the foundation for calculating the margin of error which gives us a range where the true mean is likely to fall.