Question

The owner of a large car dealership believes that the financial crisis decreased the number of customers visiting her dealership. The dealership has historically had 800 customers per day. The owner takes a sample of 100 days and finds the average number of customers visiting the dealership per day was 750. Assume that the population standard deviation is 350. At a 5% significance level, the critical value(s) is(are)

Answer 1

`z=(750-800)/((350)/(√(100)))=-1.43`

Critical Value

Since we have a lower tail test the critical value can be founded on the lower tail of the normal distribution. if we look for a value in the normal standard distribution that accumulates 0.05 of the area in the left we got:

`z_(cric)= -1.64`

Explanation:

Data given and notation

`\bar X=750` represent the sample mean

`\sigma=350` represent the population standard deviation

`n=100` sample size

`\mu_o =800` represent the value that we want to test

`\alpha=0.05` represent the significance level for the hypothesis test.

z would represent the statistic (variable of interest)

`p_v` represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean is lower than 800, the system of hypothesis would be:

Null hypothesis:
`\mu \geq 800`

Alternative hypothesis:
`\mu < 800`

Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:

`z=(\bar X-\mu_o)/((\sigma)/(√(n)))` (1)

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic

We can replace in formula (1) the info given like this:

`z=(750-800)/((350)/(√(100)))=-1.43`

Critical Value

Since we have a lower tail test the critical value can be founded on the lower tail of the normal distribution. if we look for a value in the normal standard distribution that accumulates 0.05 of the area in the left we got:

`z_(cric)= -1.64`