Cloudiness in Breslau. In the paper "Cloudiness: Note on a Novel Case of Frequency" (Proceedirgs of the Royal Society of London, Vol. 62. pp. 287-290), K. Pearson examined data on daily degree of cloudiness, on a scale of to , at Breslau (Wroclaw), Poland, during the decade . A frequency distribution of the data is presented in the following table. From the table, we find that the mean degree of cloudiness is with a standard deviation of .
a. Consider simple random samples of days during the decade in question. Approximately what percentage of such samples have a mean degree of cloudiness exceeding ?
b. Would it be reasonable to use a normal distribution to obtain the percentage required in part (a) for samples of size ? Explain your answer.
Part (b) No.
The frequency distribution of the data is depicted in the table below based on the information provided.
Given the mean degree of cloudiness is with a standard deviation of
That is and
Let denotes the number of degree of cloudiness.
A population variable has a normal distribution with a mean and standard deviation The variable is then normally distributed for samples of size , with a mean $mu$ and standard deviation
The formula used: Standard deviation
We need to figure out what percentage of day simple random samples have a mean degree of cloudiness greater than
Sampling distribution of the sample mean
Sampling distribution of the sample standard deviation
We need to figure out what proportion of the sample mean degree of cloudiness is greater than
The sample size is ; the assumption for addressing this problem is that the degree of cloudiness distribution is not normally distributed, as shown in part (A).
A statistic is said to be an unbiased estimator of a parameter if the mean of all its possible values equals the parameter; otherwise, it is said to be a biased estimator. An unbiased estimator yields, on average, the correct value of the parameter, whereas a biased estimator does not.
Part (a): Is the sample mean an unbiased estimator of the population mean? Explain your answer.
Part (b): Is the sample median an unbiased estimator of the population mean? Explain your answer.
The following graph shows the curve for a normally distributed variable. Superimposed are the curves for the sampling distributions of the sample mean for two different sample sizes.
a. Explain why all three curves are centered at the same place.
b. Which curve corresponds to the larger sample size? Explain your answer.
c. Why is the spread of each curve different?
d. Which of the two sampling-distribution curves corresponds to the sample size that will tend to produce less sampling error? Explain your answer.
c. Why are the two sampling-distribution curves normal curves?
NBA Champs The winner of the 2012-2013 National Basketball Association (NBA) championship was the Miami Heat. One possible starting lineup for that team is as follows.
a. Determine the population mean height, , of the five players:
b. Consider samples of size 2 without replacement. Use your answer to Exercise 7.11(b) on page 295 and Definition 3.11 on page 140 to find the mean, , of the variable .
c. Find using only the result of part (a).
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