Understanding Z Scores with Practical Examples

What is the z-score of sample X, if $n=36,\mathrm{\mu <!--\; \mu \; -->}=13.7,\text{St.Dev}=21,{\mathrm{\mu <!--\; \mu \; -->}}_X=9.9$?

What is the purpose of calculating z-scores?

What is the z-score of sample X, if $n=36,\mathrm{\mu <!--\; \mu \; -->}=12.3,\text{St.Dev}=47,{\mathrm{\mu <!--\; \mu \; -->}}_X=18.5$?

Supervisor Tom Choi at Tempe Equipment Company is concerned that material is not arriving as promptly as needed at work cells. A new kanban system has been installed, but there seems to be some delay in getting the material moved to the work cells so that the job can begin promptly. Choi is interested in determining how much delay there is on the part of his highly paid machinists. Ideally, the delay would be close to zero. He has asked his assistant to determine the delay factor among his 10 work cells. The assistant collects the data on a random basis over the next 2 weeks and determines that of the 1,200 observations, 100 were made while the operators were waiting for materials. Use a 99% confidence level and a 3% acceptable error.
In your report, the percent of time a delay occurs =________ % (enter your response as a percentage rounded to two decimal places). Based on the required confidence level and the acceptable error rate, the sample size of 1,200 is

What is the z-score of sample X, if $n=256,\mathrm{\mu <!--\; \mu \; -->}=17,\text{St.Dev}=80,{\mathrm{\mu <!--\; \mu \; -->}}_X=22$?

IQ scores have a mean of 100 and a standard deviation of 16. Albert Einstein reportwdly had ab IQ of 160.
a. What is the difference between Einstein's IQ and the mean?
b. How many standard deviations is that?
c. Convert Einstein's IQ score to a z-score.
d. If we consider usual IQ scores to be those that convert z scores between -2 and 2, is Einstein's IQ usual or unusual?

With a height of 55 in., Shaquille O'Neal is the tallest player on the Miami Heat basketball team. Basketball player for the Miami Heat have heights with a mean of 30.0 in. and a standard deviation of 3.5 in. Compute 33.Z-score?

What is the z-score of sample X, if ${\displaystyle n=196,\mu =45,St.Dev.=28,\text{and}E\left[X\right]=29}$

You toss a coin 163 times and get 50 heads. Test the hypothesis that it is a fair coin, using a two-tailed test and 5% level of significance.
The answer wants each of the following: test statistic, critical value, and accept or reject the hypothesis.
It is quite obvious to me that we will reject the null hypothesis since the 2 z-scores associated with getting less than 50 tails (50.5 using continuity correction) are -4.86 and 4.86, while at a 5% confidence interval, the two z-scores are -1.96 and 1.96, but I am not sure what he wants for the test statistic or critical value, or what those are exactly.

What is the z-score of sample X, if ${\displaystyle n=169,\mu =42,St.Dev.=39,\text{and}E\left[X\right]=12}$?

A sample of n = 20 scores is transformed in to z-scores. The mean for the 20 z-scores is ____?
If a researcher reports $t(18)=1.30,p>.05$ for a two-sample independent-measures experiment, the results were significant. T or F?

What is the z-score of sample X, if $n=225,\mathrm{\mu <!--\; \mu \; -->}=90,St.Dev.=90,andE[X]=78$

What is the z-score of sample X, if $n=256,\mathrm{\mu <!--\; \mu \; -->}=42,\text{St.Dev}=80,{\mathrm{\mu <!--\; \mu \; -->}}_X=89$?

The 10th grader Jack and the 11th grader Michael have identical math scores (i.e., they both got 95 on Exam 1). Thus, they also have identical z-scores.
a) True. That's what z scores are.
b) Not necessarily. Depends on the standard deviation in each of their classes.

I understood that naturally occurring phenomenon such a weight of all individuals form a bell curved normal distribution, with more frequency applied to the center. but I'm confused with the arithmetic operators used to derive the Z-score of the standard normal distribution.
Say, I found the mean weight of a population to around 60kg and to convert to standard normal distribution I first subtracted the individual weight/random measure by the mean i.e. with 60kg but then moving on why are we dividing it by (sigma) the standard deviation to find distance of how far the data point is away from the mean. Why did we use the arithmetic operation of division and why divide with sigma (the standard deviation) to find the Z-score? Can someone help out to conceptualize the idea.

In the Z-score formula, can the standard error measurement be used instead of standard deviation?

What is the z-score of X, if n = 135, ${\displaystyle \mu =70}$, SD = 1, and X =75?

Read and analyze the statements below. Write TRUE if the statement is correct, otherwise write FALSE.
1. Nearly the entire distribution (about 99.7% of the area under the curve) falls within 2 standard deviations of the mean.
2. The standard score is obtained by dividing the difference of the raw score and the mean to the standard deviation of a given normally distributed variable.
3. If $\mathrm{\mu <!--\; \mu \; -->}=95.5$, $\mathrm{\sigma <!--\; \sigma \; -->}=10$, and X = 115.5, then the z-score is 2.
4. A distribution which is not normal can be normalized by changing all the scores in the distribution into z-scores.
5. All z-scores to the right side of the curve are negative while those to the left are positive.

What is the z-score of X, if n = 4, ${\displaystyle \mu =60}$, SD =3, and X =60?

A negatively-skewed distribution has = 60, = 10. If this entire distribution is transformed into z-scores, describe the shape, mean, and standard deviation for the resulting distribution of z-scores.