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Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student's t -distribution . (Remember, use a Student's t -distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually n is large or the sample size is large).
If you are testing a single population mean , the distribution for the test is for means :
$\overline{X}~N\left({\mu}_{X},\frac{{\sigma}_{X}}{\sqrt{n}}\right)$ or ${t}_{df}$
The population parameter is μ . The estimated value (point estimate) for μ is $\overline{x}$ , the sample mean.
If you are testing a single population proportion , the distribution for the test is for proportions or percentages:
${P}^{\prime}~N\left(p,\sqrt{\frac{p\cdot q}{n}}\right)$
The population parameter is p . The estimated value (point estimate) for p is p′ . p′ = $\frac{x}{n}$ where x is the number of successes and n is the sample size.
When you perform a hypothesis test of a single population mean μ using a Student's t -distribution (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed . You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed).
When you perform a hypothesis test of a single population mean μ using a normal distribution (often called a z -test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.
When you perform a hypothesis test of a single population proportion p , you take a simple random sample from the population. You must meet the conditions for a binomial distribution which are: there are a certain number n of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success p . The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and nq must both be greater than five ( np >5 and nq >5). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with μ = p and $\sigma =\sqrt{\frac{pq}{n}}$ . Remember that q = 1 – p .
In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.
When testing for a single population mean:
When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean number of failures satisfy the conditions: np >5 and nq > n where n is the sample size, p is the probability of a success, and q is the probability of a failure.
If there is no given preconceived α , then use α = 0.05.
Which two distributions can you use for hypothesis testing for this chapter?
A normal distribution or a Student’s t -distribution
Which distribution do you use when you are testing a population mean and the standard deviation is known? Assume sample size is large.
Which distribution do you use when the standard deviation is not known and you are testing one population mean? Assume sample size is large.
Use a Student’s t -distribution
A population mean is 13. The sample mean is 12.8, and the sample standard deviation is two. The sample size is 20. What distribution should you use to perform a hypothesis test? Assume the underlying population is normal.
A population has a mean is 25 and a standard deviation of five. The sample mean is 24, and the sample size is 108. What distribution should you use to perform a hypothesis test?
a normal distribution for a single population mean
It is thought that 42% of respondents in a taste test would prefer Brand A . In a particular test of 100 people, 39% preferred Brand A . What distribution should you use to perform a hypothesis test?
You are performing a hypothesis test of a single population mean using a Student’s t -distribution. What must you assume about the distribution of the data?
It must be approximately normally distributed.
You are performing a hypothesis test of a single population mean using a Student’s t -distribution. The data are not from a simple random sample. Can you accurately perform the hypothesis test?
You are performing a hypothesis test of a single population proportion. What must be true about the quantities of np and nq ?
They must both be greater than five.
You are performing a hypothesis test of a single population proportion. You find out that np is less than five. What must you do to be able to perform a valid hypothesis test?
You are performing a hypothesis test of a single population proportion. The data come from which distribution?
binomial distribution
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