# 9.4 Rare events, the sample, decision and conclusion  (Page 3/6)

 Page 3 / 6

## Try it

It’s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows:

H 0 : p = 0.50, H a : p >0.50

α = 0.01

p -value = 0.025

Interpret the results and state a conclusion in simple, non-technical terms.

Since the p -value is greater than the established alpha value (the p -value is high), we do not reject the null hypothesis. There is not enough evidence to support It’s a Boy Genetics Labs' stated claim that their procedures improve the chances of a boy being born.

## Chapter review

When the probability of an event occurring is low, and it happens, it is called a rare event. Rare events are important to consider in hypothesis testing because they can inform your willingness not to reject or to reject a null hypothesis. To test a null hypothesis, find the p -value for the sample data and graph the results. When deciding whether or not to reject the null the hypothesis, keep these two parameters in mind:

1. α > p -value, reject the null hypothesis
2. α p -value, do not reject the null hypothesis

When do you reject the null hypothesis?

The probability of winning the grand prize at a particular carnival game is 0.005. Is the outcome of winning very likely or very unlikely?

The outcome of winning is very unlikely.

The probability of winning the grand prize at a particular carnival game is 0.005. Michele wins the grand prize. Is this considered a rare or common event? Why?

It is believed that the mean height of high school students who play basketball on the school team is 73 inches with a standard deviation of 1.8 inches. A random sample of 40 players is chosen. The sample mean was 71 inches, and the sample standard deviation was 1.5 years. Do the data support the claim that the mean height is less than 73 inches? The p -value is almost zero. State the null and alternative hypotheses and interpret the p -value.

H 0 : μ >= 73
H a : μ <73
The p -value is almost zero, which means there is sufficient data to conclude that the mean height of high school students who play basketball on the school team is less than 73 inches at the 5% level. The data do support the claim.

The mean age of graduate students at a University is at most 31 y ears with a standard deviation of two years. A random sample of 15 graduate students is taken. The sample mean is 32 years and the sample standard deviation is three years. Are the data significant at the 1% level? The p -value is 0.0264. State the null and alternative hypotheses and interpret the p -value.

Does the shaded region represent a low or a high p -value compared to a level of significance of 1%?

The shaded region shows a low p -value.

What should you do when α > p -value?

What should you do if α = p -value?

Do not reject H 0 .

If you do not reject the null hypothesis, then it must be true. Is this statement correct? State why or why not in complete sentences.

Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal.

Is this a test of means or proportions?

means

What symbol represents the random variable for this test?

In words, define the random variable for this test.

the mean time spent in jail for 26 first time convicted burglars

Is the population standard deviation known and, if so, what is it?

Calculate the following:

1. $\overline{x}$ _______
2. σ _______
3. s x _______
4. n _______

1. 3
2. 1.5
3. 1.8
4. 26

Since both σ and ${s}_{x}$ are given, which should be used? In one to two complete sentences, explain why.

State the distribution to use for the hypothesis test.

$\overline{X}~N\left(2.5,\frac{1.5}{\sqrt{26}}\right)$

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. Conduct a hypothesis test to determine if the population mean time on death row could likely be 15 years.

1. Is this a test of one mean or proportion?
2. State the null and alternative hypotheses.
H 0 : ____________________ H a : ____________________
3. Is this a right-tailed, left-tailed, or two-tailed test?
4. What symbol represents the random variable for this test?
5. In words, define the random variable for this test.
6. Is the population standard deviation known and, if so, what is it?
7. Calculate the following:
1. $\overline{x}$ = _____________
2. s = ____________
3. n = ____________
8. Which test should be used?
9. State the distribution to use for the hypothesis test.
10. Find the p -value.
11. At a pre-conceived α = 0.05, what is your:
1. Decision:
2. Reason for the decision:
3. Conclusion (write out in a complete sentence):

for poisson distribution mean............variance.
both are equal to mu
Faizan
mean=variance
Faizan
what is a variable
something that changes
Festus
why we only calculate 4 moment of mean? asked in papers.
why we only 4 moment of mean ? asked in BA exam
Faizan
Hello, can you please share the possible questions that are likely to be examined under the topic: regression and correlation analysis.
Refiloe
for normal distribution mean is 2 & variance is 4 find mu 4?
repeat quastion again
Yusuf
find mu 4. it can be wrong but want to prove how.
Faizan
for a normal distribution if mu 4 is 12 then find mu 3?
Question hi wrong ha
Tahir
ye BA mcqs me aya he teen he. 2dafa aya he
Faizan
if X is normally distributed. (n,b). then its mean deviation is?
Faizan
The answer is zero, because all odd ordered central moments of a normal distribution are Zero.
nikita
which question is zero
Faizan
sorry it is (5,16) in place of (n,b)
Faizan
I got. thanks. it is zero.
Faizan
How did we get the 24000
where do I start
in a large restaurant an average of every 7 customers ask for water with the their meal. A random sample of 12 customers is selected, find the probability that exactly 6 ask for water with their meal
any body with idea
Rufai
conditional probability
Ramesh
Rufai
iam really sorry. it's been long since I used these things. I just gave you a hint though
Ramesh
ok
Rufai
this follows binomial distribution. p(X=6)=12C6*(0.6)^6*0.4^6 use this formula n find.
syeda
can you explain the cosidered variable in the formula
Divya
x is variable wich is exactly 6 costumers
syeda
n is number of customers
syeda
ncx*p^X*q^X?
Divya
q^n-x
syeda
oh right !!! thanks yaar
Divya
I agree with Seyda too
Hoshyar
I agree with Syeda too
Hoshyar
7/12 =0.58is it?
yousaf
.
yousaf
r8
khalid
what is descriptive statistic
Descriptive statistics are brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population. Descriptive statistics are broken down into measures of central tendency and measures of variability (spread).
Divya
are you getting this ?
Divya
if so let me know
Divya
yes m getting
Ramesh
fine
Divya
what's taking place can l join u
Anest
yeah !!why not? sure
Divya
okey thanks
Anest
where are statistics used
hello
Giannis
Hi
Makhosi
how u doing
Muhid
everywhere
Faizan
the upper quartile of the population 10,12,14,16,18,20,25,15,11,11,17,is................?
Gach
The probability range is 0 to 1... but why we take it 0 to 1....
because in probability 1 means success and 0 means failure and it cnnt be more or less than 1 and 0.
syeda
b/c v hv mazimum probibliy 1 and minimum which is.no.probiblity is 0.so.v hv the range from 0 to 1
khalid
the size of a set is greeter than its subset
Hoshyar
The probability of an event will not be less than 0.  This is because 0 is impossible (sure that something will not happen).The probability of an event will not be more than 1. This is because 1 is certain that something will happen
Divya
what do they mean in a question when you are asked to find P40 and P88
Mani
hi
Mehri
you're asked to find page 40 and page 88 on that particular book.
Joseph
hi
ravi
any suggestions for statistics app better than this
ravi
sorry miss wrote the question
omar
No problem) By the way. I NEED a program For statistical data analysis. Any suggestion?
Mani
Eviews will help u
Hello
Okonkwo
arey there any data analyst and working on sas statistical model building
ravi
Hi guys ,actually I have dicovered that the P40 and P88 means finding the 40th and 88th percentiles 😌..
Megrina
who can explain the euclidian distance
ravi
I am fresh student of statistics (BS) plz guide me best app or best website relative to stat topics
Noman
IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are the mean, median, and mode.