Math / Data & Statistics

Question1. The accompanying data on flexural strength ( MPa ) for concrete beams of a certain type was introduced in Example 1.2. \begin{tabular}{rrrrrrr} 5.9 & 7.2 & 7.3 & 6.3 & 8.1 & 6.8 & 7.0 \\ 7.6 & 6.8 & 6.5 & 7.0 & 6.3 & 7.9 & 9.0 \\ 8.2 & 8.7 & 7.8 & 9.7 & 7.4 & 7.7 & 9.7 \\ 7.8 & 7.7 & 11.6 & 11.3 & 11.8 & 10.7 & \end{tabular} a. Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion, and state which estimator you used. Hint: $\quad \Sigma x_{i}=219.8$ . b. Calculate a point estimate of the strength value that separates the weakest $50 \%$ of all such beams from the strongest $50 \%$ , and state which estimator you used. c. Calculate and interpret a point estimate of the population standard deviation $\sigma$ . Which estimstor did you use? Hint: $\quad \sum x_{i}^{2}=1860.94$ . d. Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa . Hint: Think of an observation as a "success" if it exceeds 10. e. Calculate a point estimate of the population coefficient of variation $\sigma / \mu$ , and state which estimator you used.

Studdy Solution

STEP 1

1. The data set consists of 30 observations of flexural strength in MPa.
2. The sum of all observations is given as $\Sigma x_{i} = 219.8$ .
3. The sum of squares of all observations is given as $\sum x_{i}^{2} = 1860.94$ .
4. The sample size $n = 30$ .

STEP 2

1. Calculate the point estimate of the mean.
2. Calculate the point estimate of the median.
3. Calculate the point estimate of the population standard deviation.
4. Calculate the point estimate of the proportion exceeding 10 MPa.
5. Calculate the point estimate of the population coefficient of variation.

STEP 3

Calculate the mean using the formula:
$\bar{x} = \frac{\Sigma x_{i}}{n}$
Substitute the given values:
$\bar{x} = \frac{219.8}{30} = 7.3267$
The point estimate of the mean is $\bar{x} = 7.3267$ .

STEP 4

Order the data points to find the median. The data set is already provided in a tabular format, so we will arrange it in ascending order.
Ordered data: $5.9, 6.3, 6.3, 6.5, 6.8, 6.8, 7.0, 7.0, 7.2, 7.3, 7.4, 7.6, 7.7, 7.7, 7.8, 7.8, 7.9, 8.1, 8.2, 8.7, 9.0, 9.7, 9.7, 10.7, 11.3, 11.6, 11.8$
The median is the middle value. Since $n = 30$ , the median is the average of the 15th and 16th values.
$\text{Median} = \frac{7.8 + 7.8}{2} = 7.8$
The point estimate of the median is $7.8$ .

STEP 5

Calculate the standard deviation using the formula:
$s = \sqrt{\frac{\sum x_{i}^{2} - \frac{(\Sigma x_{i})^2}{n}}{n-1}}$
Substitute the given values:
$s = \sqrt{\frac{1860.94 - \frac{(219.8)^2}{30}}{29}}$
$s = \sqrt{\frac{1860.94 - 1611.2533}{29}} = \sqrt{\frac{249.6867}{29}} = \sqrt{8.6109} \approx 2.9353$
The point estimate of the population standard deviation is $s \approx 2.9353$ .

STEP 6

Count the number of observations exceeding 10 MPa. From the ordered data, these are: $10.7, 11.3, 11.6, 11.8$ .
Number of successes = 4
Proportion:
$\hat{p} = \frac{\text{Number of successes}}{n} = \frac{4}{30} = 0.1333$
The point estimate of the proportion is $\hat{p} = 0.1333$ .

STEP 7

Calculate the coefficient of variation using the formula:
$CV = \frac{s}{\bar{x}}$
Substitute the values:
$CV = \frac{2.9353}{7.3267} \approx 0.4005$
The point estimate of the population coefficient of variation is $CV \approx 0.4005$ .

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