Tell each of the following statements is true or false. Let µ represent the popu

Question

Tell each of the following statements is true or false. Let µ represent the population mean, p the population proportion, x the sample mean, and pˆ the sample proportion. Assume that all samples are drawn completely random. (3pts each)

1. The binomial random variable is a continuous random variable.

2. The probability density function can be greater than

3. The probability density function can be negative.

4. The cumulative distribution function can be greater than

5. A statistic is any quantity whose value can be calculated from sample data.

6. x is a statistic.

7. pˆ is an unbiased estimate of p.

8. When the sample size is sufficiently large, pˆ has approximately a uniform distribution.

9. When the sample size is sufficiently large, µ has approximately a normal distribution.

Explanation / Answer

1] False, binomial random variable is a descrete random variable.

let us see the example given below.

If we toss a coin two times, the statistical experiment wil have following outcomes as HH,HT,TH and TT.

so let random variable X represent number of heads, then it can take only values 0,1 or 2. so it is a descrete random variable.

2] probability function cantake the value greater than one.

for example, the uniform distribution inthe interval [0,1/2] has prob density for f(x)=2 for 0<=x<=1/2 and f(x)=0 else where.

3] false, probability density function cannot be negative. its integral over the entire space is equal to one.

4] cumulative distribution function is descrrete distribution function and random variable takes values less than x.

5] true, as statistic value can be calculated from the given set of data.

6] true, given sample mean as x.

that implies we can calculate statistic from sample data.

7] true, sample proportion is an unbiassed estimator of population proportion as determines the tendancy,for the statistics to assume values closed to the parameter.

8] true, sample proportion has uniform distribution.

9] true, population mean is normally distributed. [8,9 are since by central limit theorem]

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