SOPHIA STATISTICS MILESTONE 5 PRACTICE
QUESTIONS AND STEP-BY-STEP SOLUTIONS
2026
◉ critical value. Answer: A value that can be compared to the test
statistic to decide the outcome of a hypothesis test
◉ standard normal table. Answer: A table showing the values of the
cumulative distribution function of the standard normal
distribution.
◉ Z-Test for Population Means. Answer: A hypothesis test that
compares a hypothesized mean from the null hypothesis to a sample
mean, when the population standard deviation is known.
Step 1: State the null and alternative hypotheses.
Step 2: Check the conditions necessary in order to actually perform
the inference that you're trying to do.
Step 3: Calculate the test statistic--in this case, a z-statistic--and
calculate the p-value based on the normal sampling distribution.
Step 4: Compare your test statistic to your chosen critical value or
your p-value to our chosen significance level. Those are both
acceptable approaches. Based on how they compare, state a decision
regarding the null hypothesis. Circle it back around to the null
,hypothesis and decide if it supports the null hypothesis or refutes
the hypothesis. Make a decision to either reject or fail to reject it
based on your evidence. It should also be in the context of the
problem.
◉ hypothesis test. Answer: Step 1: State the null and alternative
hypotheses.
Step 2: Check the conditions necessary for inference.
Step 3: Calculate the test statistic and calculate the p-value based on
the normal sampling distribution.
◉ Hypothesis Test for Population Proportions. Answer: A hypothesis
test where we compare to see if the sample proportion of
"successes" differs significantly from a hypothesized value that we
believe is the population proportion of "successes."
◉ Z-test for Population Proportions. Answer: A type of hypothesis
test used to test an assumed population proportion.
Step 1: State the null and alternative hypotheses
Step 2: Check the conditions necessary for inference
Step 3: Calculate the test statistic and the p-value
Step 4: Compare your test statistic to your chosen critical value, or
your p-value to your chosen significance level. Based on how they
,compare, state a decision about the null hypothesis and conclusion
in the context of the problem.
◉ critical value. Answer: A value that can be compared to the test
statistic to decide the outcome of a hypothesis test
◉ confidence interval. Answer: An interval that contains likely
values for a parameter. We base our confidence interval on our point
estimate, and the width of the interval is affected by confidence level
and sample size.
◉ Confidence Interval for a Population Proportion. Answer: A
confidence interval that gives a likely range for the value of a
population proportion. It is the sample proportion, plus and minus
the margin of error from the normal distribution.
◉ t-distribution. Answer: A family of distributions that are centered
at zero and symmetric like the standard normal distribution, but
heavier in the tails. Depending on the sample size, it does not
diminish towards the tails as fast. If the sample size is large, the t-
distribution approximates the normal distribution.
◉ t-test for population. Answer: State the null and alternative
hypotheses.
Check the conditions necessary for inference.
, Calculate the test statistic and its p-value.
Compare our test statistic to our chosen critical value, or our p-
value, to the significance level, and then based on how those
compare, make a decision about the null hypothesis and a
conclusion in the context of the problem.
◉ T-test For Population Means. Answer: The type of hypothesis test
used to test an assumed population mean when the population
standard deviation is unknown. Due to the increased variability in
using the sample standard deviation instead of the population
standard deviation, the t-distribution is used in place of the z-
distribution.
◉ confidence interval. Answer: An interval we are some percent
certain (eg 90%, 95%, or 99%) will contain the population
parameter, given the value of our sample statistic.
◉ t distribution. Answer: A distribution similar to the normal
distribution but with fatter tails. Depending on the sample size, it
does not diminish toward the tails as fast.
Step 1: Verify the conditions necessary for inference.
Step 2: Calculate the confidence interval.
Step 3: Interpret the confidence interval.
QUESTIONS AND STEP-BY-STEP SOLUTIONS
2026
◉ critical value. Answer: A value that can be compared to the test
statistic to decide the outcome of a hypothesis test
◉ standard normal table. Answer: A table showing the values of the
cumulative distribution function of the standard normal
distribution.
◉ Z-Test for Population Means. Answer: A hypothesis test that
compares a hypothesized mean from the null hypothesis to a sample
mean, when the population standard deviation is known.
Step 1: State the null and alternative hypotheses.
Step 2: Check the conditions necessary in order to actually perform
the inference that you're trying to do.
Step 3: Calculate the test statistic--in this case, a z-statistic--and
calculate the p-value based on the normal sampling distribution.
Step 4: Compare your test statistic to your chosen critical value or
your p-value to our chosen significance level. Those are both
acceptable approaches. Based on how they compare, state a decision
regarding the null hypothesis. Circle it back around to the null
,hypothesis and decide if it supports the null hypothesis or refutes
the hypothesis. Make a decision to either reject or fail to reject it
based on your evidence. It should also be in the context of the
problem.
◉ hypothesis test. Answer: Step 1: State the null and alternative
hypotheses.
Step 2: Check the conditions necessary for inference.
Step 3: Calculate the test statistic and calculate the p-value based on
the normal sampling distribution.
◉ Hypothesis Test for Population Proportions. Answer: A hypothesis
test where we compare to see if the sample proportion of
"successes" differs significantly from a hypothesized value that we
believe is the population proportion of "successes."
◉ Z-test for Population Proportions. Answer: A type of hypothesis
test used to test an assumed population proportion.
Step 1: State the null and alternative hypotheses
Step 2: Check the conditions necessary for inference
Step 3: Calculate the test statistic and the p-value
Step 4: Compare your test statistic to your chosen critical value, or
your p-value to your chosen significance level. Based on how they
,compare, state a decision about the null hypothesis and conclusion
in the context of the problem.
◉ critical value. Answer: A value that can be compared to the test
statistic to decide the outcome of a hypothesis test
◉ confidence interval. Answer: An interval that contains likely
values for a parameter. We base our confidence interval on our point
estimate, and the width of the interval is affected by confidence level
and sample size.
◉ Confidence Interval for a Population Proportion. Answer: A
confidence interval that gives a likely range for the value of a
population proportion. It is the sample proportion, plus and minus
the margin of error from the normal distribution.
◉ t-distribution. Answer: A family of distributions that are centered
at zero and symmetric like the standard normal distribution, but
heavier in the tails. Depending on the sample size, it does not
diminish towards the tails as fast. If the sample size is large, the t-
distribution approximates the normal distribution.
◉ t-test for population. Answer: State the null and alternative
hypotheses.
Check the conditions necessary for inference.
, Calculate the test statistic and its p-value.
Compare our test statistic to our chosen critical value, or our p-
value, to the significance level, and then based on how those
compare, make a decision about the null hypothesis and a
conclusion in the context of the problem.
◉ T-test For Population Means. Answer: The type of hypothesis test
used to test an assumed population mean when the population
standard deviation is unknown. Due to the increased variability in
using the sample standard deviation instead of the population
standard deviation, the t-distribution is used in place of the z-
distribution.
◉ confidence interval. Answer: An interval we are some percent
certain (eg 90%, 95%, or 99%) will contain the population
parameter, given the value of our sample statistic.
◉ t distribution. Answer: A distribution similar to the normal
distribution but with fatter tails. Depending on the sample size, it
does not diminish toward the tails as fast.
Step 1: Verify the conditions necessary for inference.
Step 2: Calculate the confidence interval.
Step 3: Interpret the confidence interval.
