SOPHIA STATISTICS MILESTONE 5 KEY
CONCEPTS AND REVIEW NOTES 2026
◉ sample mean. Answer: A mean obtained from a sample of a given
size. Denoted as x bar.
◉ population parameters. Answer: Summary values for the
population. These are often unknown.
◉ population mean. Answer: A mean for all values in the population.
Denoted as μ.
◉ sampling with replacement. Answer: A sampling plan where each
observation that is sampled is replaced after each time it is sampled,
resulting in an observation being able to be selected more than once.
◉ sampling error. Answer: The amount by which the sample statistic
differs from the population parameter.
◉ sample size. Answer: The size of a sample of a population of
interest.
,◉ distribution of sample means. Answer: Step 1: First, take these
sample means and graph them. Draw out an axis. For this one, it
should go from 1 to 4 because this set can't average anything higher
than four or lower than a one.
Step 2: Take the average value, for example, the mean of 2.5, and put
a dot at 2.5 on the x-axis, much like a dot plot. Do this for all the
sample means that you have found.
Step 3: You can keep doing this over and over again. Ideally, you
would do this hundreds or thousands of times, to show the
distribution of all possible samples that could be taken of size four.
Once you've enumerated every possible sample of size four from this
spinner, then the sampling distribution looks like this:
A distribution where each data point consists of a mean of a
collected sample. For a given sample size, every possible sample
mean will be plotted in the distribution.
◉ Standard Deviation of a Distribution of Sample Means. Answer:
The standard deviation of the population, divided by the square root
of sample size.
◉ standard error. Answer: The standard deviation of the sampling
distribution of sample means.
◉ central limit theorem. Answer: A theorem that explains the shape
of a sampling distribution of sample means. It states that if the
,sample size is large (generally n ≥ 30), and the standard deviation of
the population is finite, then the distribution of sample means will
be approximately normal.
◉ Distribution of Sample Proportions. Answer: The distribution of
all possible sample proportions for a certain size, n.
◉ hypothesis testing. Answer: The standard procedure in statistics
for testing claims about population parameters.
◉ hypothesis. Answer: A claim about a population parameter.
◉ null hypothesis. Answer: A claim about a particular value of a
population parameter that serves as the starting assumption for a
hypothesis test.
◉ alternative hypothesis. Answer: A claim that a population
parameter differs from the value claimed in the null hypothesis.
◉ statistical significance. Answer: The statistic obtained is so
different from the hypothesized value that we are unable to attribute
the difference to chance variation.
, ◉ practical significance. Answer: An arbitrary assessment of
whether observations reflect a practical real-world use.
◉ type 1 error. Answer: An error that occurs when a true null
hypothesis is rejected.
◉ type 2 error. Answer: An error that occurs when a false null
hypothesis is not rejected.
◉ significance level. Answer: The probability of making a Type I
error. Abbreviated with the symbol, alpha.
◉ Power of a Hypothesis Test. Answer: The probability that we
reject the null hypothesis (correctly) when a difference truly does
exist.
◉ one-tailed test. Answer: A test for when you have reason to
believe the population parameter is higher or lower than the
assumed parameter value of the null hypothesis.
◉ right-tailed test. Answer: A hypothesis test where the alternative
hypothesis only states that the parameter is higher than the stated
value from the null hypothesis.
CONCEPTS AND REVIEW NOTES 2026
◉ sample mean. Answer: A mean obtained from a sample of a given
size. Denoted as x bar.
◉ population parameters. Answer: Summary values for the
population. These are often unknown.
◉ population mean. Answer: A mean for all values in the population.
Denoted as μ.
◉ sampling with replacement. Answer: A sampling plan where each
observation that is sampled is replaced after each time it is sampled,
resulting in an observation being able to be selected more than once.
◉ sampling error. Answer: The amount by which the sample statistic
differs from the population parameter.
◉ sample size. Answer: The size of a sample of a population of
interest.
,◉ distribution of sample means. Answer: Step 1: First, take these
sample means and graph them. Draw out an axis. For this one, it
should go from 1 to 4 because this set can't average anything higher
than four or lower than a one.
Step 2: Take the average value, for example, the mean of 2.5, and put
a dot at 2.5 on the x-axis, much like a dot plot. Do this for all the
sample means that you have found.
Step 3: You can keep doing this over and over again. Ideally, you
would do this hundreds or thousands of times, to show the
distribution of all possible samples that could be taken of size four.
Once you've enumerated every possible sample of size four from this
spinner, then the sampling distribution looks like this:
A distribution where each data point consists of a mean of a
collected sample. For a given sample size, every possible sample
mean will be plotted in the distribution.
◉ Standard Deviation of a Distribution of Sample Means. Answer:
The standard deviation of the population, divided by the square root
of sample size.
◉ standard error. Answer: The standard deviation of the sampling
distribution of sample means.
◉ central limit theorem. Answer: A theorem that explains the shape
of a sampling distribution of sample means. It states that if the
,sample size is large (generally n ≥ 30), and the standard deviation of
the population is finite, then the distribution of sample means will
be approximately normal.
◉ Distribution of Sample Proportions. Answer: The distribution of
all possible sample proportions for a certain size, n.
◉ hypothesis testing. Answer: The standard procedure in statistics
for testing claims about population parameters.
◉ hypothesis. Answer: A claim about a population parameter.
◉ null hypothesis. Answer: A claim about a particular value of a
population parameter that serves as the starting assumption for a
hypothesis test.
◉ alternative hypothesis. Answer: A claim that a population
parameter differs from the value claimed in the null hypothesis.
◉ statistical significance. Answer: The statistic obtained is so
different from the hypothesized value that we are unable to attribute
the difference to chance variation.
, ◉ practical significance. Answer: An arbitrary assessment of
whether observations reflect a practical real-world use.
◉ type 1 error. Answer: An error that occurs when a true null
hypothesis is rejected.
◉ type 2 error. Answer: An error that occurs when a false null
hypothesis is not rejected.
◉ significance level. Answer: The probability of making a Type I
error. Abbreviated with the symbol, alpha.
◉ Power of a Hypothesis Test. Answer: The probability that we
reject the null hypothesis (correctly) when a difference truly does
exist.
◉ one-tailed test. Answer: A test for when you have reason to
believe the population parameter is higher or lower than the
assumed parameter value of the null hypothesis.
◉ right-tailed test. Answer: A hypothesis test where the alternative
hypothesis only states that the parameter is higher than the stated
value from the null hypothesis.
