The central limit theorem states that:a.The mean of a population is equal to the mean of the sampleb.The sample mean will always be equal to the population meanc.The sampling distribution of the sample mean approaches a normal distribution as the sample size increasesd.The sample mean is always greater than the population mean
Question
The central limit theorem states that:a.The mean of a population is equal to the mean of the sampleb.The sample mean will always be equal to the population meanc.The sampling distribution of the sample mean approaches a normal distribution as the sample size increasesd.The sample mean is always greater than the population mean
Solution 1
The central limit theorem states that:
a. The mean of a population is equal to the mean of the sample. b. The sample mean will always be equal to the population mean. c. The sampling distribution of the sample mean approaches a normal distribution as the sample size increases. d. The sample mean is always greater than the population mean.
Step 1: Understand the central limit theorem. The central limit theorem is a fundamental concept in statistics that relates to the behavior of sample means. It states that as the sample size increases, the distribution of sample means approaches a normal distribution, regardless of the shape of the population distribution.
Step 2: Understand the first statement. The first statement, "The mean of a population is equal to the mean of the sample," means that the average value of a population is equal to the average value of a sample taken from that population. This is a basic principle in statistics.
Step 3: Understand the second statement. The second statement, "The sample mean will always be equal to the population mean," means that if you calculate the mean of a sample, it will be equal to the mean of the entire population. This assumes that the sample is representative of the population.
Step 4: Understand the third statement. The third statement, "The sampling distribution of the sample mean approaches a normal distribution as the sample size increases," is a key concept in the central limit theorem. It means that as the sample size gets larger, the distribution of sample means becomes more and more like a normal distribution. This is true regardless of the shape of the population distribution.
Step 5: Understand the fourth statement. The fourth statement, "The sample mean is always greater than the population mean," is incorrect. The sample mean can be greater than, equal to, or less than the population mean. It depends on the specific values in the sample and the population.
In summary, the central limit theorem states that the mean of a population is equal to the mean of the sample, the sample mean will always be equal to the population mean, the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, and the sample mean can be greater than, equal to, or less than the population mean.
Solution 2
The correct answer is c. The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases. This theorem is a key foundation of many statistical procedures. It does not state that the mean of a population is equal to the mean of the sample (option a), that the sample mean will always be equal to the population mean (option b), or that the sample mean is always greater than the population mean (option d).
Similar Questions
According to the Central Limit Theorem, the sampling distribution of the sample mean becomes approximately normally distributed asthe standard error increasesthe population variance decreasesthe size of the population increasesthe number of samples drawn increasesthe size of the sample increases
The central limit theorem states that if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean:Group of answer choicesis approximately normal if n ≥ 30.is approximately normal if the underlying population is normal.has the same variance as the population.is approximately normal if n < 30.
Central Limit Theorem for sampling distribution is valid only when:Select one:a. Large samples drawn from any independent and identically distributed populationb. The population follows normal distributionc. Samples are drawn from any independent and identically distributed populationd. The population follows a normal distribution and the sample size should be at least 30
Now consider the Central Limit Theorem (CLT). How confidently can you use a sample from this population to make inferences about the population mean?
Why is the Central Limit Theorem so convenient?Question 1Answera.Because of that we know that the mean will also be in the center.b.Because we know how likely sample-means will be.c.a & bClear my choiceQuestion 2Not yet answeredMarked out of 1.00Flag questionTipsQuestion textThe Central Limit Theorem states that every distribution is always normally distributedQuestion 2Answera.Trueb.FalseClear my choiceQuestion 3Not yet answeredMarked out of 1.00Flag questionTipsQuestion textAn important condition for the central limit theorem is for the sample size to be sufficiently large. What is the minimum sample size for the sample to be considered sufficiently large?Question 3Answera.1b.100c.30Clear my choiceQuestion 4Not yet answeredMarked out of 1.00Flag questionTipsQuestion textFrom the distribution of a random variable X a random sample of size n is drawn. What is the distribution of x̄?A. normally distributedB. x̄ )C. a and bQuestion 4Answera.Ab.Bc.CClear my choiceQuestion 5Not yet answeredMarked out of 1.00Flag questionTipsQuestion textWhich requirements should the Sampling Distribution fulfill to be normally distributed?Question 5Answera.Population must be normally distributed.b.Sample size must be sufficiently large.c.You must have more than 30 observations.d.b & cClear my choice
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