Question

A psychologist wants to know whether memory performance is reduced by old age. She randomly selects 67 elderly individuals and finds that their mean score on a standardized memory test equals 514. Scores on the standardized memory test in the general population are distributed normally with a mean equal to 600 and a standard deviation equal to 112. Is there sufficient evidence at the 0.1 significance level to conclude that memory performance is reduced by old age

Answer 1

`z=(514-600)/((112)/(√(67)))=-6.285`

The p value for this case is given by:

`p_v =P(z<-6.285)=1.64x10^(-10)`

Since the p value is very low than the significance level given we have enough evidence to conclude that the true mean for the scores on a standardized memory is significantly lower than 600 and then we can conclude that memory performance is reduced by old age

Explanation:

Information given

`\bar X=514` represent the sample mean for the scores on a standardized memory

`\sigma=112` represent the population standard deviation

`n=67` sample size

`\mu_o =600` represent the value that we want to verify

`\alpha=0.1` represent the significance level

z would represent the statistic

`p_v` represent the p value

System of hypothesis

We want to check if the true mean for this case is less than 600, the system of hypothesis would be:

Null hypothesis:
`\mu \geq 600`

Alternative hypothesis:
`\mu < 600`

Since we know the population deviation the statistic for this case is given by:

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

Replacing the info given we got:

`z=(514-600)/((112)/(√(67)))=-6.285`

The p value for this case is given by:

`p_v =P(z<-6.285)=1.64x10^(-10)`

Since the p value is very low than the significance level given we have enough evidence to conclude that the true mean for the scores on a standardized memory is significantly lower than 600 and then we can conclude that memory performance is reduced by old age