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What else can I help you with?
Is the mean for a set of data always greater than the standard deviation?
Yes; the standard deviation is the square root of the mean, so it will always be larger.
What is the relationship between the relative size of the starndard deviation and the kurtosis of a distribution?
It is inversely proportional; a larger standard deviation produces a small kurtosis (smaller peak, more spread out data) and a smaller standard deviation produces a larger kurtosis (larger peak, data more centrally located).
How does a sample size impact the standard deviation?
If I take 10 items (a small sample) from a population and calculate the standard deviation, then I take 100 items (larger sample), and calculate the standard deviation, how will my statistics change? The smaller sample could have a higher, lower or about equal the standard deviation of the larger sample. It's also possible that the smaller sample could be, by chance, closer to the standard deviation of the population. However, A properly taken larger sample will, in general, be a more reliable estimate of the standard deviation of the population than a smaller one. There are mathematical equations to show this, that in the long run, larger samples provide better estimates. This is generally but not always true. If your population is changing as you are collecting data, then a very large sample may not be representative as it takes time to collect.
True or False As the amount of confidence increases the required sample size should decrease Explain your answer?
I can examine this as a question of theory or real life: As a matter of theory, I will rephrase your question as follows: Does theoretical confidence interval of the mean (CI) of a sample, size n become larger as n is reduced? The answer is true. This is established from the sampling distribution of the mean. The sampling distribution is the probability distribution of the mean of a sample, size n. I will also consider the question as a matter of real life: If I take a sample from a population, size 50 and calculate the CI and take a smaller sample, say size 10, will I calculate a larger CI? If I use the standard deviation calculated from the sample, this is not necessarily true. The CI should be larger but I can't say in every case it will belarger. The standard deviation of the sample will vary from sample to sample. I hope this answers your question. You can find more information on confidence intervals at: http://onlinestatbook.com/chapter8/mean.html
Does the standard deviation of x decrease in magnitude as the size of the sample gets smaller?
No. But a small sample will be a less accurate predictor of the standard deviation of the population due to its size. Another way of saying this: Small samples have more variability of results, sometimes estimates are too high and other times too low. As the sample size gets larger, there's a better chance that your sample will be close to the actual standard deviation of the population.
When to t test z test t interval z interval etc?
Use a t-test when comparing the means of two groups, especially when the sample size is small (n < 30) and the population standard deviation is unknown. A z-test is appropriate for large sample sizes (n ≥ 30) or when the population standard deviation is known. For confidence intervals, use a t-interval for smaller samples with unknown population standard deviation, and a z-interval for larger samples or known population standard deviation. Always check if the data meets the assumptions for each test before proceeding.
Can standard deviation be larger then its variance?
No. The standard deviation is the square root of the variance.
What happen to the width of a confidence interval if the sample size is doubled from 100 to 200?
When the sample size is doubled from 100 to 200, the width of the confidence interval generally decreases. This occurs because a larger sample size reduces the standard error, which is the variability of the sample mean. As the standard error decreases, the margin of error for the confidence interval also decreases, resulting in a narrower interval. Thus, a larger sample size leads to more precise estimates of the population parameter.
What affect does increasing the sample size have on the width of the confidence interval?
Increasing the sample size decreases the width of the confidence interval. This occurs because a larger sample provides more information about the population, leading to a more accurate estimate of the parameter. As the sample size increases, the standard error decreases, which results in a narrower interval around the sample estimate. Consequently, the confidence interval becomes more precise.
The mean of A is 14 with a standard deviation of 4.2. The mean of B is 16 with a standard deviation of 4.4 Which is more dispersed?
B because the spread, in this case standard deviation, is larger.
What standard deviation is larger 8.2 or 10.8?
10.8
How do you produce a new confidence interval with a larger width?
Increase your percent confidence to provide an increased width.
How can you decrease the width of a confidence interval without sacrificing the level of confidence?
To decrease the width of a confidence interval without sacrificing the level of confidence, you can increase the sample size. A larger sample provides more information about the population, which reduces the standard error and narrows the interval. Additionally, using a more precise measurement technique can also help achieve a narrower interval. However, it's important to note that increasing the sample size is the most effective method for maintaining the desired confidence level while reducing width.
Is the mean for a set of data always greater than the standard deviation?
Yes; the standard deviation is the square root of the mean, so it will always be larger.
What if I have a very high standard deviation?
The larger the value of the standard deviation, the more the data values are scattered and the less accurate any results are likely to be.
What is the relationship between the relative size of the starndard deviation and the kurtosis of a distribution?
It is inversely proportional; a larger standard deviation produces a small kurtosis (smaller peak, more spread out data) and a smaller standard deviation produces a larger kurtosis (larger peak, data more centrally located).
What happens to the confidence interval when you increase the sample size?
When you increase the sample size, the confidence interval typically becomes narrower. This occurs because a larger sample size reduces the standard error, leading to more precise estimates of the population parameter. As a result, while the confidence level remains the same, the interval reflects increased certainty about the estimate. However, the actual confidence level (e.g., 95%) does not change; it simply provides a tighter range around the estimate.
