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Hello everyone.
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In this question we have three statements.
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The statement one states that for any sample size of 110, the population mean the mean of the sampling distribution for the sample means and the sample means will always be equal to each other.
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We need to check whether the statement is true or false.
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Similarly, in the second statement we have, a large sample size will result in a smaller sample standard deviation for the sample.
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We need to check whether this statement is true or false.
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Finally, in the third statement we have, if the sample size is 100 and the population standard deviation is 20, then the standard deviation of the sampling distribution for mean is 2.
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We need to check whether this statement is true or false.
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Recall if x1, x2 till x n be n independent and identically distributed, which is given by iid samples from a population with mean mu and standard deviation sigma.
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So the sample size is n.
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Then we define the sample mean as x bar which is a random variable and is given as x1 plus x2 plus the summation going till x n divided by the sample size n let us denote it as equation one now the mean of the sampling distribution for the sample means or in other words the expected value of the sample mean is denoted as e x bar and according to definition this is equal to the population mean.
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This is denoted by equation 2.
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Now let us analyze the statement 1.
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In statement 1 we have for any sample size of 110, the population mean, the mean of the sampling distribution for this sample means and the sample mean will always be equal to each other.
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Need to check whether this statement is true or false.
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So for statement 1 we have the sample size.
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Let us denote it as n and it is equal to 110 according to the statement 1.
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Now the mean of the sampling distribution for the sample means which is denoted by ex is equal to mu and mu is the population mean according to equation two.
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Now both these quantities are independent of the sample size n.
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Therefore in statement one we can write the mean of the sampling distribution for the sample means denoted as e x bar will be equal to the population mean mu.
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However, the sample mean that is given by x bar according to equation 1 we can write as x1 plus x2 plus the summation goes on till x subscript 110 as the sample size is 110 and this is divided by the 2...
