A preliminary investigation of a site leads an engineer to state that the relative weights are 3 to 5 to 2 (respectively) that the unconfined compressive strength of the soil below is 1200, 1000, or 800 psf (the only three values considered possible, for simplicity). "Undisturbed" samples of the soil will be obtained by boring and tested to gain further information. Owing to the difficulties in obtaining such samples and owing to testing inaccuracies, the following frequencies of indicated strengths are considered applicable for each specimen: P[indicated strength | state] State Indicated strength 800 1000 1200 800 0.8 0.4 0.2 1000 0.2 0.5 0.3 1200 0.0 0.1 0.5 1.0 1.0 1.0 The engineer calls for a sampling plan of two independent specimens. (a) Find the conditional probabilities of each of the possible outcomes of this sample of size two given that the true strength is 1200 psf. (b) If the results of the sampling were one specimen indicating 1000 and one indicating 800 psf, find the engineer's posterior probabilities of the strength. (c) Suppose that after these two specimens the engineer continued sampling and found an uninterrupted sequence of specimens indicating 1200 psf. After how many could he stop: (i) Confident that the strength was not actually 800? (ii) At least "90 percent confident" that the strength was actually 1200? Note: In your solutions, please use the following notation: T_i = { True strength is i }; Z_i^j = { strength indicated by specimen j is i }
Added by Eva A.
Close
Step 1
From the given table, we know that P(Z_1000 | T_1200) = 0.1 and P(Z_800 | T_1200) = 0.0. Since the specimens are independent, we can multiply the probabilities: P(Z_1000, Z_800 | T_1200) = P(Z_1000 | T_1200) * P(Z_800 | T_1200) = 0.1 * 0.0 = 0.0 Similarly, Show more…
Show all steps
Your feedback will help us improve your experience
Ameer Said and 50 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
In order to ensure the quality of concrete material used in reinforced concrete construction, concrete cylinders are collected at random from concrete mixes delivered to the construction site by a mixing plant. Past records of concrete from the same plant show that 90% of concrete mixes are good or of satisfactory quality. To further ensure that the concrete delivered on site is of good quality, the engineer requires that one cylinder among those collected each day be tested for minimum compressive strength. The test method is not perfect—its reliability is only 97%, meaning the probability that a good-quality concrete cylinder will pass the test is 0.97, or that a poor-quality cylinder can pass the test is 0.03. How would you calculate the probability that out of the 5000 concrete cylinders to be used in the construction project, 200 of them will turn out to be of bad quality?
Sri K.
Shear strength measurements derived from unconfined compression tests for two types of soil gave the results shown in the following table (measurements in tons per square foot). Soil Type I Soil Type II n1 = 30 n2 = 35 y1 = 1.69 y2 = 1.46 s1 = 0.24 s2 = 0.22 Do the soils appear to differ with respect to average shear strength, at the 1% significance level? State the null and alternative hypotheses. H0: μ1 = μ2 Ha: μ1 ≠ μ2 State the rejection region. (Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.) z > ___ z < ___ Calculate the appropriate test statistic. (Round your answer to two decimal places. Calculate your test statistic using y1 - y2.) z = What is the conclusion of your test? Fail to reject H0. There is not enough evidence to conclude the mean shear strength for the two soil types is different.
Kari H.
(i) What is the level of significance? State the null and alternate hypotheses. (ii) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? Compute the corresponding $z$ or $t$ value as appropriate. (iii) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (iv) Based on your answers in parts (i)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (v) Interpret your conclusion in the context of the application.Note: For degrees of freedom $d . f$. not in the Student's $t$ table, use the closest $d . f$. that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount, and therefore produce a slightly more "conservative" answer.Answers may vary due to rounding. Bell Peppers The pathogen Phytophthora capsici causes bell peppers to wilt and die. Because bell peppers are an important commercial crop, this organism has undergone a great deal of agricultural research. It is thought that too much water aids the spread of the pathogen. Two fields are under study. The first step in the research project is to compare the mean soil water content for the two fields (Source: Journal of Agricultural, Biological, and Environmental Statistics, Vol. 2, No. 2). Units are percentage of water by volume of soil.Field A samples, $x_{1}:$ $$\begin{array}{llllll}10.2 & 10.7 & 15.5 & 10.4 & 9.9 & 10.0 \\ 15.1 & 15.2 & 13.8 & 14.1 & 11.4 & 11.5\end{array}$$.Field B samples, $x_{2}:$$$\begin{array}{rrrrrrrr}8.1 & 8.5 & 8.4 & 7.3 & 8.0 & 7.1 & 13.9 & 12.2 \\13.4 & 11.3 & 12.6 & 12.6 & 12.7 & 12.4 & 11.3 & 12.5\end{array}$$.Use a calculator with mean and standard deviation keys to verify that $\bar{x}_{1} \approx 12.53, s_{1} \approx 2.39$ $\bar{x}_{2} \approx 10.77,$ and $s_{2} \approx 2.40$ (a) Assuming the distribution of soil water content in each field is mound-shaped and symmetrical, use a $5 \%$ level of significance to test the claim that field $A$ has, on average, a higher soil water content than field B. (b) Find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$. Explain the meaning of the confidence interval in the context of the problem.
Chi-Square and $F$ Distributions
Chi-Square: Goodness of Fit
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD