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The mean number of minutes for app engagement by a tablet user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of 60.

  1. What are the mean and standard deviation for the sample mean number of app engagement by a tablet user?
  2. What is the standard error of the mean?
  3. Find the 90 th percentile for the sample mean time for app engagement for a tablet user. Interpret this value in a complete sentence.
  4. Find the probability that the sample mean is between eight minutes and 8.5 minutes.
  1. μ x ¯ = μ = 8.2   σ x ¯ = σ n = 1 60 = 0.13
  2. This allows us to calculate the probability of sample means of a particular distance from the mean, in repeated samples of size 60.
  3. Let k = the 90 th percentile
    k = invNorm ( 0. 90,8 .2, 1 60 ) = 8.37. This values indicates that 90 percent of the average app engagement time for table users is less than 8.37 minutes.
  4. P (8< x ¯ <8.5) = normalcdf ( 8,8 .5,8 .2, 1 60 ) = 0.9293
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Try it

Cans of a cola beverage claim to contain 16 ounces. The amounts in a sample are measured and the statistics are n = 34, x ¯ = 16.01 ounces. If the cans are filled so that μ = 16.00 ounces (as labeled) and σ = 0.143 ounces, find the probability that a sample of 34 cans will have an average amount greater than 16.01 ounces. Do the results suggest that cans are filled with an amount greater than 16 ounces?

We have P (( x ¯ >16.01) = normalcdf ( 16 .01,E99,16, 0.143 34 ) = 0.3417. Since there is a 34.17% probability that the average sample weight is greater than 16.01 ounces, we should be skeptical of the company’s claimed volume. If I am a consumer, I should be glad that I am probably receiving free cola. If I am the manufacturer, I need to determine if my bottling processes are outside of acceptable limits.

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References

Baran, Daya. “20 Percent of Americans Have Never Used Email.”WebGuild, 2010. Available online at http://www.webguild.org/20080519/20-percent-of-americans-have-never-used-email (accessed May 17, 2013).

Data from The Flurry Blog, 2013. Available online at http://blog.flurry.com (accessed May 17, 2013).

Data from the United States Department of Agriculture.

Chapter review

In a population whose distribution may be known or unknown, if the size ( n ) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size ( n ).

Formula review

The Central Limit Theorem for Sample Means: X ¯ ~ N ( μ x σ x n )

The Mean X ¯ : μ x

Central Limit Theorem for Sample Means z-score and standard error of the mean: z = x ¯ μ x ( σ x n )

Standard Error of the Mean (Standard Deviation ( X ¯ )): σ x n

Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let X ¯ be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.

What is the mean, standard deviation, and sample size?

mean = 4 hours; standard deviation = 1.2 hours; sample size = 16

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Complete the distributions.

  1. X ~ _____(_____,_____)
  2. X ¯ ~ _____(_____,_____)

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Find the probability that one review will take Yoonie from 3.5 to 4.25 hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

  1. This is a frequency curve for a normal distribution. It shows a single peak in the center with the curve tapering down to the horizontal axis on each side. The distribution is symmetrical. The horizontal axis represents the random variable X.
  2. P (________< x <________) = _______

a. Check student's solution.
b. 3.5, 4.25, 0.2441

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Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

  1. This is a frequency curve for a normal distribution. It shows a single peak in the center with the curve tapering down to the horizontal axis on each side. The distribution is symmetrical. The horizontal axis represents the random variable X.
  2. P (________________) = _______

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What causes the probabilities in [link] and [link] to be different?

The fact that the two distributions are different accounts for the different probabilities.

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Find the 95 th percentile for the mean time to complete one month's reviews. Sketch the graph.

  1. This is a frequency curve for a normal distribution. It shows a single peak in the center with the curve tapering down to the horizontal axis on each side. The distribution is symmetrical. The horizontal axis represents the random variable X.
  2. The 95 th Percentile =____________
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Questions & Answers

how do you find z if you only know the area of .0808
Cady Reply
How to take a random sample of 30 observations
Hamna Reply
you can use the random function to generate 30 numbers or observation
smita
How we can calculate chi-square if observed x٫y٫z/frequency 40,30,20 Total/90
Insha Reply
calculate chi-square if observed x,y,z frequency 40,30,20total 90
Insha
find t value,if boysN1, ،32,M1,87.43 S1square,39.40.GirlsN2,34,M2,82.58S2square,40.80 Determine whether the results are significant or insignificant
Insha
The heights of a random sample of 100 entering HRM Freshman of a certain college is 157 cm with a standard deviation of 8cm. test the data against the claim that the overall height of all entering HRM students is 160 cm. previous studies showed that
Crispen Reply
complete the question.. as data given N = 100,mean= 157 cm, std dev = 8 cm..
smita
Z=x-mu/ std dev
smita
find the mean of 25,26,23,25,45,45,58,58,50,25
Asmat Reply
add all n divide by 10 i.e 38
smita
38
hhaa
amit
1 . The “average increase” for all NASDAQ stocks is the:
Jamshaid Reply
STATISTICS IN PRACTICE: This is a group assignment that seeks to reveal students understanding of statistics in general and it’s practical usefulness. The following are the guidelines; 1.      Each group has to identify a natural process or activity and gather data about/from the process. 2.     
Kofi Reply
The diameter of an electric cable,say, X is assumed to be continoues random variable with p.d.f f(x)=6x(1-x); ≤x≤1 a)check that f(X) is p.d.f b) determine a number b such that p(Xb)
Syed Reply
A manufacturer estimate 3% of his output is defective. Find the probability that in a sample of 10 items (a) less than two will be defective (b) more than two will be defective.
ISAIAH Reply
A manufacturer estimates that 3% of his output of a small item is defective. Find the probabilities that in a sample of 10 items (a) less than two and (b) more than two items will be defective.
ISAIAH
use binomial distribution with parameter n=10, p= 0.03, q=0.97
Shivprasad
the standard deviation of a symmetrical distribution is 7.8 . what must be the value of forth moment about the mean in order that distribution be a) leptokurtic b) mesokurtic c) platy kyrtic intrept the obtain value of a b and c
Tushar Reply
A researcher observed that four out of every ten of their products are normally defective. A total of 360 samples of the products were being tested. If the sample is normally distributed and 220 of the products were identified to be faulty, test the hypothesis that the observation of the res
Adepoju Reply
false
HariPrasad
please answer the ques"following values are obtained from life table T15=3,493,601 and e°15=44.6 then expected number of person alive at exact age 15 will be "
vinay
make it clear
Kagimu
how x minus x bar is equal to zero
Kashif Reply
When the mean (X bar) of the sample and the datapoint-in-context (X) from the same sample are the same, then it (X minus X bar) is equal to 0
Johns
e.g. mean of. sample is 3 and one of the datapoints in that sample is also 3
Johns
a numerical value used as a summary measure for a sample such as a sample mean is known as
rana Reply
differentiate between qualitative and quantitative variables
rana Reply
qualitative variables are descriptive while quantitative are numeric variables
Chisomo
please guys what is the formulas use in calculated statistics please iam new here
Yunisa Reply
Dear Yunisa there are different formulas used in statistics depending on wnat you want to measure. It would be helpful if you can be more specific
LAMIN

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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