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Sampling distributions

Resource managers of forest game lands were concerned about the size of the deer and rabbit populations during the winter months in a particular forest. As an estimate of population size, they proposed using the average number of pellet groups for rabbits and deer per 30-foot-square plots. From an aerial photograph, the forest was divided into N= 10,000 30-foot-square grids. A simple random sample of 2 = 500 plots was taken, and the number of pellet groups was observed for rabbits and for deer. The results of this study are summarized in the accompanying table. a. Estimate $$\displaystyle\mu_{{1}},{\quad\text{and}\quad}\mu_{{2}}$$, the average number of pellet groups for deer and rabbits, respectively, per 30-foot-square-plots. Place bounds on the errors of estimation. b. Estimate the difference in the mean size of pellet groups per plot for the two animals, with an appropriate margin of error. Deer Sample mean = 2.30 Sample variance = 0.65 Rabbits Sample mean = 4.52 Sample variance = 0.97

Sampling distributions

Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article ''Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis'' (Intl. J. of Advanced Manuf. Tech., 2010: 117–134)). If $$\bar{X}$$ is the sample mean Young’s modulus for a random sample of $$n = 16$$ sheets, where is the sampling distribution of $$\bar{X}$$ centered, and what is the standard deviation of the $$\bar{X}$$ distribution?

Sampling distributions

In government data, a house-hold consists of all occupants of a dwelling unit, while a family consists of 2 or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States. $$\text{Number of people}\\ \begin{array}{|l|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \begin{array}{l} \text { Household } \\ \text { probability } \end{array} & 0.25 & 0.32 & 0.17 & 0.15 & 0.07 & 0.03 & 0.01 \\ \hline \begin{array}{l} \text { Family } \\ \text { probability } \end{array} & 0 & 0.42 & 0.23 & 0.21 & 0.09 & 0.03 & 0.02 \\ \hline \end{array}$$ Let H = the number of people in a randomly selected U.S. household and F = the number of people in a randomly chosen U.S. family. The standard deviations of the 2 random variables are $$\sigma H=1.421$$ and $$\sigma F=1.249.$$ Explain why this difference makes sense.

Sampling distributions

Describe how to use random numbers to simulate the following PERFORMANCES: (a) A basketball player has the ABILITY to make 40% of his shots and he takes 25 shots in a game. (b) A basketball player has the ABILITY to make 75.2% of her free-throws and she takes 8 free throws in a game.

Sampling distributions

Consider the two samples of data from the McKenzie School. The numbers represent the time in seconds that it took each child to cover a distance of 50 meters. Girls’ Times: 8.3, 8.6, 9.5, 9.5, 9.6, 9.8, 9.9, 9.9, 10.0, 10.0, 10.0, 10.1, 10.3, 10.5 Boys’ Times: 7.9, 8.0, 8.2, 8.2, 8.4, 8.6, 8.8, 9.1, 9.3, 9.5, 9.8, 9.8, 10.0, 10.1, 10.3. Based on the sample means, do you conclude that the distributions of times from the boys’ population and girls’ population are different? Explain.

Sampling distributions

Represent sampling distributions in the format of a table that lists the different values of the sample statistic along with their corresponding probabilities. Given that the data consist of ranks, does it really make sense to identify the sampling distribution of the sample means?

Sampling distributions

Test the hypothesis using the P-value approach. Be sure to verify the requirements of the test. $$H_0:\ P=0.52$$ versus $$H_1:p<0.52$$ $$N=150, X=72,\alpha=0.1$$ Is $$NP_0(1-P_0)$$ greater than or equal to 10? Use technology to find the P-Value.

Sampling distributions

When comparing two sets of data values, what is the advantage of using relative frequency distributions instead of frequency distributions?

Sampling distributions

Use the row of numbers shown below to generate 12 random numbers between 01 and 99. 78038 18022 84755 23146 12720 70910 49732 79606 Starting at the beginning of the row, what are the first 12 numbers between 01 and 99 in the sample?

Sampling distributions

Solve by using the strategy Work Backward. A beetle needs to climb out of a crater that is 800 cm deep. It advances 120 cm each day, but is slips back 90 cm while resting each night. How many days will it take before this beetle successfully climbs out of the crater?

Sampling distributions

Carol and Alina play soccer for a local college team. Based on anecdotal evidence, they think that there is a difference in a player success rate of taking penalty kicks with their dominant foot compared to their non-dominant foot. They would like to test this hypothesis with an experiment. Carol arranges for each of the 11 starting players on her team to take ten penalty kicks with their dominant foot and ten penalty kicks with their non dominant foot and records the data. Using the same data values, describe two distributions that would be more supportive of the hypothesis.

Sampling distributions

The following data represent soil water content for independent random samples of soil taken from two experimental fields growing bell peppers Soil water content from field I: $$x_1;n_1$$=72 15.111.210.310.816.68.39.112.39.114.3 10.716.110.215.28.99.59.611.31411.3 15.611.213.89.08.48.21213.911.616 9.611.48.48.014.110.913.213.814.610.2 11.513.114.712.510.211.811.012.710.310.8 11.012.610.89.611.510.611.710.19.79.7 11.29.810.311.99.711.310.4121110.7 8.811.1 Soil water content from field II: $$x_2;n_2$$= 80 12.110.213.68.113.57.811.87.78.19.2 14.18.913.97.512.67.314.912.27.68.9 13.98.413.47.112.47.69.9267.37.4 14.38.413.27.311.37.59.712.36.97.6 13.87.513.38.011.36.87.411.711.87.7 12.67.713.213.910.412.87.610.710.710.9 12.511.310.713.28.912.97.79.79.711.4 11.913.49.213.48.811.97.18.51414.2 Which distribution (standard normal or Student's t) did you use? Why? Do you need information about the soil water content distributions?

Sampling distributions

1) Describe sampling distributions and sampling variavility 2) Explain The Central Limit Theorem 3) Explain how confidence intervals are created and what can they tell us about population parameters

Sampling distributions

The correct statement which is incorrect from the options about the sampling distribution of the sample mean (a) the standard deviation of the sampling distribution will decrease as the sample size increases, (b) the standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples, (c) the sample mean is an unbiased estimator of the true population mean, (d) the sampling distribution shows how the sample mean will vary in repeated samples, (e) the sampling distributions shows how the sample was distributed around the sample mean.

Sampling distributions

Which of the following is true about the sampling distribution of means? A. Shape of the sampling distribution of means is always the same shape as the population distribution, no matter what the sample size is. B. Sampling distributions of means are always nearly normal. C. Sampling distributions of means get closer to normality as the sample size increases. D. Sampling distribution of the mean is always right skewed since means cannot be smaller than 0.

Sampling distributions

Which of the following statements about the sampling distribution of the sample mean is incorrect? (a) The standard deviation of the sampling distribution will decrease as the sample size increases. (b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples. (c) The sample mean is an unbiased estimator of the population mean. (d) The sampling distribution shows how the sample mean will vary in repeated samples. (e) The sampling distribution shows how the sample was distributed around the sample mean.

Sampling distributions

Explain whether the central limit theorem can be applied and assert that the sampling distributions of A and Bare approximately normal, if the sample sizes of A and Bare large.

Sampling distributions

Explain the importance of the statement "Sampling distributions play a key role in the process of statistical interference" stated by the researchers Turner and Dabney.

Sampling distributions