Hypothesis Testing (Normal Distribution) (CIE A Level Maths: Probability & Statistics 2)

The random variable  follows a normal distribution with a mean of 40 and a standard deviation of 8.  The mean of 16 independent observations of  is denoted as .

Explain why the standard deviation of  is 2.

By standardising and using the table of values for the normal distribution, find:

By standardising and using the table of critical values for the normal distribution, find:

the value of  such that 

The population mean of the random variable   is being tested using a null hypothesis  against the alternative hypothesis A random sample of 10 observations is taken from the population and the sample mean is calculated as .

Write down the distribution of the sample mean,. 

Find the test statistic, , for  and hence find .

The population mean of the random variable  is being tested using a null hypothesis against an alternative hypothesis.  A random sample of 36 observations is taken from the population and the critical region for the test is .

Write down the distribution of the sample mean, . 

A random sample of size 100 is taken from a population given by .

A two-tailed test is used to investigate the null hypothesis  at the 10% level of significance.

Find the rejection regions for  for this test.

Given that there is insufficient evidence to reject the null hypothesis when  write down an inequality for the range of values of

Write suitable null and alternative hypotheses for each of the following situations.

The Starlighter is a new brand of flashlight.  It is known that the brightness,  lumens, of the light emitted from a Starlighter follows a normal distribution with a standard deviation of 15 lumens.  Annie, a salesperson, claims that the mean brightness of a Starlighter is greater than 110 lumens.  To test her claim, the null hypothesis  is used with a 5% level of significance.

Write down a suitable alternative hypothesis to test Annie’s claim.

To test Annie’s claim a random sample of 40 Starlighters is taken and the mean brightness, , is calculated.

Assuming that the null hypothesis is true write down the distribution of the sample mean, .

Given that the mean of sample is lumens,

The wingspan of a small white butterfly,  cm, follows a normal distribution with a standard deviation of 0.8 cm.  A report states that the average wingspan of a small white butterfly is 4.1 cm.  Kenzie, a butterfly enthusiast, wants to conduct a two-tailed hypothesis test, using a 5% level of significance, to investigate the validity of the statement made by the report.

Kenzie uses a random sample of 6 small white butterflies and finds that the mean wingspan is 2.65 cm.  Kenzie starts off the hypothesis test as follows:

Identify and explain the two mistakes that Kenzie has made in his hypothesis test.

A hypothesis test is used to investigate the population mean of the random variable . A random sample of size 16 is used to test the null hypothesis .

Write down the probability of a Type I error if a 10% significance level is used.

Find the probability of a Type I error given that the rejection region is

Given that the rejection region is , find the probability of a Type II error if the true mean is 27.