Sampling Distributions - AP Statistics

A t test is used to compare the means of different groups. A t test score describes the likelihood that the difference in means between two groups is due to chance. The null hypothesis assumes the two sets are equal however, one can reject the null hypothesis with a p value within a particular confidence level.

The population standard deviation =

The sample mean standard deviation =

There are two keys here. One, we have a large sample size since , meaning we can use the Central Limit Theorem even if points per game is not normally distributed.

Our -score thus becomes...

where is the specified points or less needed this season,

is the average points per game of the previous season,

is the standard deivation of the previous season,

and is the number of games.

Use the fact that .

Alternately, you can use the fact that the variance of the sample mean varies inversely by the square root of the sample size, so to reduce the variance by a factor of 10, the sample size needs to be 100.

A sample size of bags means that the central limit theorem is applicable and the distribution can be assumed to be normal. The sample mean would be and

Therefore,

The correct answer is the distribution of average height statistics that could happen from all possible samples of college students. Remember that a sampling distribution isn't just a statistic you get form taking a sample, and isn't just a piece of data you get from doing sampling. Instead, a sampling distribution is a distribution of sample statistics you could get from all of the possible samples you might take from a given population.

We can infer that roughly 68% of random samples of college students will have a sample mean of between 65 and 75 inches. Anytime we try to make an inference from a sampling distribution, we have to keep in mind that the sampling distribution is a distribution of samples and not a distribution about the thing we're trying to measure itself (in this case the height of college students). Also, remember that the empirical rules tells us that roughly 68% of the distribution will fall within one standard deviation of the mean.

The standard error (SE) is the standard deviation of the sampling distribution.

First check to see if the Central Limit Theorem applies. Since n > 30, it does. Next we need to calculate the standard error. To do that we divide the population standard deviation by the square-root of n, which gives us a standard error of 0.646. Next, we calculate a z-score using our z-score formula:

Plugging in gives us:

Finally, we look up our z-score in our z-score table to get a p-value.

The table gives us a p-value of,

The correct answer is the distribution of average height statistics that could happen from all possible samples of college students. Remember that a sampling distribution isn't just a statistic you get form taking a sample, and isn't just a piece of data you get from doing sampling. Instead, a sampling distribution is a distribution of sample statistics you could get from all of the possible samples you might take from a given population.

The population standard deviation =

The sample mean standard deviation =

There are two keys here. One, we have a large sample size since , meaning we can use the Central Limit Theorem even if points per game is not normally distributed.

Our -score thus becomes...

where is the specified points or less needed this season,

is the average points per game of the previous season,

is the standard deivation of the previous season,

and is the number of games.

Use the fact that .

Alternately, you can use the fact that the variance of the sample mean varies inversely by the square root of the sample size, so to reduce the variance by a factor of 10, the sample size needs to be 100.

A sample size of bags means that the central limit theorem is applicable and the distribution can be assumed to be normal. The sample mean would be and

Therefore,

We can infer that roughly 68% of random samples of college students will have a sample mean of between 65 and 75 inches. Anytime we try to make an inference from a sampling distribution, we have to keep in mind that the sampling distribution is a distribution of samples and not a distribution about the thing we're trying to measure itself (in this case the height of college students). Also, remember that the empirical rules tells us that roughly 68% of the distribution will fall within one standard deviation of the mean.

The standard error (SE) is the standard deviation of the sampling distribution.

First check to see if the Central Limit Theorem applies. Since n > 30, it does. Next we need to calculate the standard error. To do that we divide the population standard deviation by the square-root of n, which gives us a standard error of 0.646. Next, we calculate a z-score using our z-score formula:

Plugging in gives us:

Finally, we look up our z-score in our z-score table to get a p-value.

The table gives us a p-value of,

A t test is used to compare the means of different groups. A t test score describes the likelihood that the difference in means between two groups is due to chance. The null hypothesis assumes the two sets are equal however, one can reject the null hypothesis with a p value within a particular confidence level.

Use the fact that the sum of squared standard and independent normal variables follows a chi-squared distribution with degrees of freedom.

This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. Therefore, the null hypothesis is that the relationship is not significant, meaning the slope of the line is equal to zero.