z-scores for sample means Flashcards
What is the probability that I pick an individual at random from the population with IQ > 106?
Population mean = 100
Population SD = 15
z = (x - pop. mean) / pop. SD
z = (106 - 100) / 15 = 0.40
z = 0.40
look at the p-value table and find 0.4 ABOVE population mean = 0.3446 = 34.46%
Answer = The probability of picking an individual at random from the population with IQ > 106 is 34.46%
I take a random sample of 25 people from IQ distribution. What is the probability that I get a sample mean IQ > 106?
Pop. mean = 100
Pop. SD = 15
Sample mean = 100
Sample SD = ??
z score (for SDM) = (x - SDM mean) / SDM SD
SDM SD = pop. SD / SQRT (sample size)
SDM SD = 15 / SQRT (25)
z score (for SDM) = (106 - 100) / 3 = 2.00
look at the p-value table and find 2.00 ABOVE SDM = 0.0228 = 2.28%
Answer = The probability of getting a sample mean IQ > 106 is 2.28%
Assume human height (h) in cm follows a normal distribution with a population mean 170 and standard deviation 14, i.e height h~ N(170, 14).
What is the standard error of the mean for samples of size 16 from this distribution?
a. 14 b. 3.5 c. 0.875 d. 42.5
S.E. = pop. SD / SQRT (sample size)
S.E. = 14 / SQRT (16)
S.E. = 3.5
Answer = B
Assume human height (h) in cm follows a normal distribution with population mean 170 and standard deviation 14, i.e height h~ N(170, 14). You calcalute the mean height of 173cm for a sample of 16 people from this distribution.
Calculate the z-score (to 2 d.p.) associated with this sample mean
a. 0.86 b. 0.21 c. 3.43 d. 0.07
SDM SD = pop SD / SQRT (sample size)
SDM SD = 14 / SQRT (16) = 3.5
z score (SDM) = (x - SDM mean) / SDM SD
z score (SDM) = (173 - 170) / 3.5 = 0.86
Answer = A
Assume human height (h) in cm follows a normal distribution with population mean 125 and standard deviation 10, i.e height h~ N(125, 10). What is the probability of selecting a sample of 25 people from this distribution with mean height less than 123 (use your table for the area under the SND)?
a. 0.4207 b. 0.5793 c. 0.8413 d. 0.1587
SDM SD = pop. SD / SQRT (sample size)
SDM SD = 10 / SQRT (25) = 2
z score (SDM) = (x - SDM mean) / SDM SD
z score (SDM) = (123 - 125) / 2 = -1.00
Refer to the p-value table for -1.00 BELOW THE MEAN = 0.1587
Answer = D
Assume human height (h) in cm follows a normal distribution with population mean 170 and standard deviation 14, i.e height h~ N(170, 14).
What is the probability (to 4 d.p.) of selecting a sample of 25 people from this population with mean height in the range from
171cm < m < 172cm (round z scores to 2 d.p and use your table for area under the SND)?
a. 0.9300 b. 0.8795 c. 0.1205 d. 0.0700
SDM SD = pop. SD / SQRT (sample size)
SDM SD = 14 / SQRT (25) = 2.8
z score (SDM) = (x - SDM mean) / SDM SD
z score (SDM) = (171 - 170) / 2.8 = 0.36
z score (SDM) = (172 - 170) / 2.8 = 0.71
Refer to the p-value table for 0.36 BELOW THE MEAN = 0.6406 and 0.71 ABOVE THE MEAN = 0.2389
1 - 0.6406 - 0.2389 = 0.1205 = 12.05%
Answer = C
