A sample of n=25 scores has a mean of M=68 Find the z-score for this sample: a. If it was obtain

step 1 Okay, so we have five scores, 8, 4, 10, 0, and 3. First part is compute the mean and the standar

A sample of n=25 scores has a mean of M=68 Find the z-score...

Key Concepts

Population Parameters

Population parameters, such as the population mean and population standard deviation, are constants that describe the entire population. These values are used to compute standardized scores and to understand the variation within the population, forming a basis for inferential statistics.

Z-Score

A z-score (or standard score) indicates how many standard deviations an observation (or statistic) is away from the mean of the distribution. It provides a way to standardize different scores, enabling comparisons across different scales or distributions.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, typically the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size, and it quantifies the variability of the sample mean from sample to sample.

Sampling Distribution of the Mean

The sampling distribution of the mean refers to the probability distribution of the sample means obtained from all possible samples of a fixed size drawn from the population. According to the Central Limit Theorem, even if the population distribution is not normal, the distribution of the sample mean will tend to be normal if the sample size is sufficiently large.

Transcript

00:02 Okay, so we have five scores, 8, 4, 10, 0, and 3.

00:13 First part is compute the mean and the standard deviation.

00:16 So the mean represented, this is a sample mean, we'll call it x bar, is add up all the numbers and divide by 5, or 5 numbers, and if you do that you get a sample mean is 5.

00:37 For the standard deviation, which we'll use the label s of x, it's the following formula.

00:44 You take each individual x value, you subtract the mean from it, square it, and then you divide it by n minus 1 since this is a sample standard deviation.

00:56 This is the average distance from the mean, so we're subtracting it from the mean to figure out its distance from the mean, and then we square it, and then to finish that off and finish the problem we take the square root to compensate for the fact that we squared it.

01:10 That's how we calculate the standard deviation.

01:12 So if we go through that and do that with this problem, we're going to end up with the following.

01:17 We take the first data value minus the mean, square it, plus the second data value minus the mean, square it, so forth and so forth with every number.

01:34 And then once we've done that we have to divide it by n minus 1.

01:38 There's this one degree of freedom that we have to account for since it's a sample mean.

01:43 So in this case we're going to divide this whole thing by 4 and then take the square root of all.

01:51 So what i got when i did everything underneath the square root, i got a sum of 16 for this whole numerator, and then 16 divided by 4, we get the square root of 4 or 2.

02:08 So our sample standard deviation is 2 in this case.

02:13 Next, once we've calculated that, they want us to now find the z -score for each of the sample values.

02:22 And to do that we have the following formula.

02:24 We take each individual value, which we'll label as x, subtract the mean, and divide it by the standard deviation.

02:33 Okay, and so that's the formula for calculating a z -score.

02:37 So if we're doing that for 8, we would end up with 8 minus 5 over 2, which gives us an answer of 3 halves or 1 .5.

02:47 If we do that for the next value, which is 4, 4 minus 5 over 2, we're going to get negative 1 half or negative 0 .5.

03:01 For the next one, we're going to do it for 10, 10 minus 5 over 2, which is 5 halves, 2 .5...

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