Suppose that a population consists of only 5 members and the values of are as follows. Member 1 2 3 4 5 Y 5 3 5 2 2 Verify that Ybar is an unbiased estimator of μ.
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Suppose that a population consists of only 5 members and the values of are as follows.
| Member | 1 | 2 | 3 | 4 | 5 |
| Y | 5 | 3 | 5 | 2 | 2 |
- Verify that Ybar is an unbiased estimator of μ.
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- If X1, X2, and X3 constitute a random sample of sizen = 3 from a Bernoulli population, show that Y =X1 + 2X2 + X3 is not a sufficient estimator of θ. (Hint:Consider special values of X1, X2, and X3.)If X1, X2, and X3 constitute a random sample of sizen = 3 from a normal population with the mean μ and thevariance σ2, find the efficiency of X1 + 2X2 + X34relative toX1 + X2 + X33as estimates of μ.Suppose a research study uses two independent groups with 21 individuals in each group. If a t-test analysis produces a calculated value of t of 1.68, which of the following is the correct decision?A. At the .05 level of significance, reject the null hypothesisB. At the .01 level of significance, reject the null hypothesisC. At the .05 level of significance, fail to reject the null hypothesisD. Impossible to answer without additional information
- The number of contaminating particles on a silicon waferprior to a certain rinsing process was determined for eachwafer in a sample of size 100, resulting in the followingfrequencies:Number of particles 0 1 2 3 4 5 6 7Frequency 1 2 3 12 11 15 18 10Number of particles 8 9 10 11 12 13 14Frequency 12 4 5 3 1 2 1a. What proportion of the sampled wafers had at leastone particle? At least five particles?b. What proportion of the sampled wafers had betweenfive and ten particles, inclusive? Strictly between fiveand ten particles?c. Draw a histogram using relative frequency on thevertical axis. How would you describe the shape of thehistogram?Suppose we have three independent groups: X1, X2 and X3, and the population standard deviations are known as σ1 = σ2 = σ3 = 10. For each group, we observe 50 samples, and x ̄1 = 200, x ̄2 = 190, and x ̄3 = 210. Construct a 90% confidence interval for μ1 + μ2 + μ3.Suppose that a random sample of size 1 is to be taken from a finite population of size N. a. How many possible samples are there?b. Identify the relationship between the possible sample means and the possible observations of the variable under consideration.c. What is the difference between taking a random sample of size 1 from a population and selecting a member at random from the population?
- A sample of men and women who had passed their driver's test either the first time or the second time were surveyed, with the following results: Results of the driving testGender First time Second timeMen 126 211Women 135 178a) Do these data suggest that there is a relationship between gender and the passing of their driver’s test from which the present sample was drawn? Let alpha=.05Suppose you randomly select a sample of 16 observations from a population with �=11 attached in ss below thanks for help agoajgr rA researcher randomly selects 1000 death certificates and, after interviewing the attending physician, records the following information about the deceased: at the significance of the level of 5%, do these data show that smoking and cause of death are independent? Do you hold any assumptions here? If so, are they satisfied?
- Let X1, …, Xn be a random sample from a N(μ, σ²) population. Find the MLEs of μ and of σ.If X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.Recent surveys have shown a decline in the percentage of U.S. households withcable TV. In one city, a simple random sample of 468 households had 117households reporting they had cable TV, and 351 households reporting they didnot.(a) Show the “plug-in” box to represent our best guess for the relative frequency ofpeople with and without cable. (Let 1 correspond to with cable and 0 without).(b) The percentage of households in the city with cable TV is estimated to be___________%, and this estimate is likely to be off by about ___________% or so.(c) A 90%-confidence interval for the percentage of households in the city withcable TV goes from ___________% to ___________%.