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

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

Step-by-step explanation:

Information given

[tex]\bar X=514[/tex] represent the sample mean for the scores on a standardized memory

[tex]\sigma=112[/tex] represent the population standard deviation

[tex]n=67[/tex] sample size

[tex]\mu_o =600[/tex] represent the value that we want to verify

[tex]\alpha=0.1[/tex] represent the significance level

z would represent the statistic

[tex]p_v[/tex] 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:[tex]\mu \geq 600[/tex]

Alternative hypothesis:[tex]\mu < 600[/tex]

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

[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)

Replacing the info given we got:

[tex]z=\frac{514-600}{\frac{112}{\sqrt{67}}}=-6.285[/tex]

The p value for this case is given by:

[tex]p_v =P(z<-6.285)=1.64x10^{-10}[/tex]

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