The statistic used to estimate the mean of a population, μ, is the sample mean, . Show
If X has a distribution with mean μ, and standard deviation σ, and is approximately normally distributed or n is large, then is approximately normally distributed with mean μ and standard error..When σ Is Known If the standard deviation, σ, is known, we can transform to an approximately standard normal variable, Z:
Example: From the previous example, μ=20, and σ=5. Suppose we draw a sample of size n=16 from this population and want to know how likely we are to see a sample average greater than 22, that is P( > 22)?So the probability that the sample mean will be >22 is the probability that Z is > 1.6 We use the Z table to determine this: P( > 22) = P(Z > 1.6) = 0.0548. Exercise: Suppose we were to select a sample of size 49 in the example above instead of n=16. How will this affect the standard error of the mean? How do you think this will affect the probability that the sample mean will be >22? Use the Z table to determine the probability. Answer When σ Is Unknown If the standard deviation, σ, is unknown, we cannot transform to standard normal. However, we can estimate σ using the sample standard deviation, s, and transform to a variable with a similar distribution, the t distribution. There are actually many t distributions, indexed by degrees of freedom (df). As the degrees of freedom increase, the t distribution approaches the standard normal distribution.
If X is approximately normally distributed, then has a t distribution with (n-1) degrees of freedom (df) Using the t-tableNote: If n is large, then t is approximately normally distributed.
The z table gives detailed correspondences of P(Z>z) for values of z from 0 to 3, by .01 (0.00, 0.01, 0.02, 0.03,…2.99. 3.00). The (one-tailed) probabilities are inside the table, and the critical values of z are in the first column and top row. The t-table is presented differently, with separate rows for each df, with columns representing the two-tailed probability, and with the critical value in the inside of the table. The t-table also provides much less detail; all the information in the z-table is summarized in the last row of the t-table, indexed by df = ∞. So, if we look at the last row for z=1.96 and follow up to the top row, we find that P(|Z| > 1.96) = 0.05 Exercise: What is the critical value associated with a two-tailed probability of 0.01? Answer Now, suppose that we want to know the probability that Z is more extreme than 2.00. The t-table gives us P(|Z| > 1.96) = 0.05 And P(|Z| > 2.326) = 0.02 So, all we can say is that P(|Z| > 2.00) is between 2% and 5%, probably closer to 5%! Using the z-table, we found that it was exactly 4.56%. Example: In the previous example we drew a sample of n=16 from a population with μ=20 and σ=5. We found that the probability that the sample mean is greater than 22 is P( > 22) = 0.0548. Suppose that is unknown and we need to use s to estimate it. We find that s = 4. Then we calculate t, which follows a t-distribution with df = (n-1) = 24.From the tables we see that the two-tailed probability is between 0.01 and 0.05. P(|T| > 1.711) = 0.05 And P(|T| > 2.064) = 0.01 To obtain the one-tailed probability, divide the two-tailed probability by 2. P(T > 1.711) = ½ P(|T| > 1.711) = ½(0.05) = 0.025 And P(T > 2.064) = ½ P(|T| > 2.064) = ½(0.01) = 0.005 So the probability that the sample mean is greater than 22 is between 0.005 and 0.025 (or between 0.5% and 2.5%) Exercise: . If μ=15, s=6, and n=16, what is the probability that >18 ?Answer return to top | previous page | next page How does sample size affect the standard deviation of the sample means?Thus as the sample size increases, the standard deviation of the means decreases; and as the sample size decreases, the standard deviation of the sample means increases.
How does sample size affect mean standard deviation and standard error of the mean?Standard error increases when standard deviation, i.e. the variance of the population, increases. Standard error decreases when sample size increases – as the sample size gets closer to the true size of the population, the sample means cluster more and more around the true population mean.
How does increasing sample size affect mean and standard deviation?Therefore, as a sample size increases, the sample mean and standard deviation will be closer in value to the population mean μ and standard deviation σ .
Does the change in sample size affect the mean and standard deviation of the sampling distribution of p hat?The Sampling Distribution of the Sample Proportion
Since the sample size n appears in the denominator of the square root, the standard deviation does decrease as sample size increases. Finally, the shape of the distribution of p-hat will be approximately normal as long as the sample size n is large enough.
Does higher sample size affect standard deviation?The standard error is also inversely proportional to the sample size; the larger the sample size, the smaller the standard error because the statistic will approach the actual value. The standard error is considered part of inferential statistics. It represents the standard deviation of the mean within a dataset.
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