simple random sampling problems and solutions

First verify that the sample is sufficiently large to use the normal distribution. A population has mean $1,542$ and standard deviation $246$. Find the probability that the mean of a sample of size $64$ will be less than $46.7$. A state public health department wishes to investigate the effectiveness of a campaign against smoking. You may assume that the normal distribution applies. Is there strong evidence that people are keeping their cars longer than was the case five years ago? 5. How strong is the evidence that the campaign to reduce smoking has been effective? Let $X_1$, $X_2$, $X_3$, $X_4$ be a random sample from the $Uniform(0,1)$ distribution, and let $X_{(1)}$, $X_{(2)}$, $X_{(3)}$ , $X_{(4)}$. The random variables $X_1$, $X_2$, $X_3$, $...$, $X_n$ defined above are independent and identically distributed (i.i.d.) $\mathrm{Var}(\overline{X})=\frac{\sigma^2}{n}$. Find the probability that, when a sample of size $1,500$ is drawn from a population in which the true proportion is $0.22$, the sample proportion will be no larger than the value you computed in part (a). �q��(��GE;Rdd��Q9$��d�i��v��xpY��׽�z��?�z�zǞ�F c��L�{]��u^D��X֭�_��[͡X6�#�;�H�BF��Y��V��X g�(��18ʑ��x^�l L��G�2�ƃ)�t��}��@�s��N��_ڎԸo�Ӆ�2��k~�q�8u'��8��c��eo�?��w�p�s� ��d��.��\E=ew�Vݑ�u4�n.j�i��4l7�_HT땸�#�U� ��ݞ��a�S]gi�r �xn�4:�#}]y�x�^��Xo�Tit�w'o[�E�!�1�}(�#|Į��dc�KCĉj��/A[Ij��r�7]_�Ax��&V�8�8�[�V�q��ZEn^�؈��e��{��W^��Х�i��Ź�p�vȴ=��}�2��:��ЛK�HM�͕>5��+)b��k.��^xe��ˠ��Hﲙ�� K]�7��+{%�Y`+}q��P�5�Sc��4�Ќ�}�4�E�d��g�{# JY��(�d��[��}R�iA V��W�Z �h�v�[��}�m�K�P��Q�\ If you're seeing this message, it means we're having trouble loading external resources on our website. Find the probability that the score $X$ on a randomly selected exam paper is between $70$ and $80$. %�� ThoughtCo uses cookies to provide you with a great user experience. Find the mean and standard deviation of the sample proportion $\widehat{P}$ obtained from random samples of size $1,600$. Decide whether or not the sample size is large enough to assume that the sample proportion $\widehat{P}$ is normally distributed. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. It can require a sample size that is too large. In a population in which the true proportion is $22\%$ the chance that a random sample of size $1,500$ would produce a sample proportion of $18.6\%$ or less is only $7/100$ of $1\%$. Random samples of size $n$ produced sample proportions $\hat{p}$ as shown. Give an interpretation of the result in part (b). \end{align} \lim_{n \rightarrow \infty} P(Z_n \leq x)=\Phi(x), \qquad \textrm{ for all }x \in \mathbb{R} By. \end{align} Find the probability that in a random sample of $600$ homes, between $80\%$ and $90\%$ will have a functional smoke detector. Square the result. \begin{align}%\label{} Of them, $132$ are ten years old or older. This sample is of size 5, and odd number, so the middle value of 124 is the sample median. $\mu _{\bar{X}}=100,\; \sigma _{\bar{X}}=1.33$. Find the probability that in a random sample of $250$ men at least $10\%$ will suffer some form of color blindness. A normally distributed population has mean $25.6$ and standard deviation $3.3$. These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang. The main benefit of the simple random sample is that each member of the population has an equal chance of being chosen for the study.This means that it guarantees that the sample chosen is representative of the population and … A population has mean $48.4$ and standard deviation $6.3$. Find the probability that if you buy one such tire, it will last only $57,000$ or fewer miles. Find the probability that when he enters the restaurant today it will be at least $5$ minutes until he is served. Find the probability that a single randomly selected element $X$ of the population is between $57,000$ and $58,000$. Find the mean and standard deviation of the sample proportion $\widehat{P}$. To determine the value of $X_2$, again we choose a person uniformly (and independently from the first person) at random and let $X_2$ be the height of that person. ��B�r�h(7|7��Y��_�:�C.��ZdE1unۃs��;�p�T�~V�1��uGu�h+�x�CR ?a8a Assuming the airline’s claim is true, find the probability of a sample of size $30$ producing a sample proportion so low as was observed in this sample. Find the value of his deposit after 4 years. Then the CDF and PDF of $X_{(i)}$ are given by, Also, the joint PDF of $X_{(1)}, X_{(2)}, \cdots, X_{(n)}$ is given by. Khan Academy is a 501(c)(3) nonprofit organization. Thus the proportion of times a three is observed in a large number of tosses is expected to be close to $1/6$ or $0.1\bar{6}$. For each number: subtract the mean. stream Give an interpretation of the result in part (b). In practice, we often do the sampling without replacement, that is, we do not allow one person to be chosen twice. Suppose the sample proportion $0.15$ came from rolling the die $2,400$ times instead of only $240$ times. /Filter /FlateDecode Suppose lifetimes are normally distributed with standard deviation $\sigma =3,500$ miles. Z_n=\frac{\overline{X}-\mu}{\sigma / \sqrt{n}}=\frac{X_1+X_2+...+X_n-n\mu}{\sqrt{n} \sigma} %�� You may assume that the normal distribution applies. \end{align}. Find the probability that the mean amount of credit card debt in a sample of $1,600$ such households will be within $\$300$ of the population mean. (9 - 7), Calculate the mean of the squared differences. 2. Calculate the mean (simple average of the numbers). Some countries allow individual packages of prepackaged goods to weigh less than what is stated on the package, subject to certain conditions, such as the average of all packages being the stated weight or greater. Suppose this proportion is valid for all homes. Chapter 7: Hypothesis Testing - Solutions 7.1 Introduction to Hypothesis Testing The problem with applying the techniques learned in Chapter 5 is that typically, the popula-tion mean ( ) and standard deviation (˙) are not known. A population has mean $12$ and standard deviation $1.5$. >> For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Example 1: A school has 650 students. Suppose the mean length of time that a caller is placed on hold when telephoning a customer service center is $23.8$ seconds, with standard deviation $4.6$ seconds. Find the probability that average time until he is served in eight randomly selected visits to the restaurant will be at least $5$ minutes. Techniques for generating a simple random sample, Techniques for random sampling and avoiding bias, Types of studies (experimental vs. observational). Our mission is to provide a free, world-class education to anyone, anywhere. xڭYK�ܸ�ϯh��I�zX,��l.I�s`�A��L+VKcI=��X�Z�ጝ�^ZY,��"[nn6r�Ӆ��?_^��6*Z�dsy��&��\V�O�_��h��.�L��H��z��-O\n�$��8��]s��^��HD��]��;��"��Z;��z��D{�/�E�[�X��w[�C]��4`n�;��m�r��}=xw-��D�L'��~<5c}��D� �h�52�v�_h�)�֋�.��%��B^� lyy�,Db�rO�N4X9N��w0~29B�9�ӈlc�g��[ ��,�D�d�௖fۚ��.�_^|�P��0��E�l�ǋO��M�a0I�o��qcb��ſ|ܭ�V�2+#RP��ߝ�gI��ԓ��[� The three will be selected by simple random sampling. Result Drug (c.1) Placebo (c.2) Total Nausea (r.1) 36 13 49 Simple Random Sample: An Overview . /Filter /FlateDecode Find the probability that the mean amount of cholesterol in a sample of $144$ eggs will be within $2$ milligrams of the population mean. If a biologist induces a state of tonic immobility in such a shark in order to study it, find the probability that the shark will remain in this state for between $10$ and $13$ minutes. and we refer to them collectively as a (simple) random sample. f_X(x)=1, \qquad \textrm{ for }x \in [0,1], Find the probability that a fair die would produce a proportion of $0.15$ or less. 3 0 obj << Find the mean and standard deviation of the sample proportion $\widehat{P}$, The proportion of a population with a characteristic of interest is $p = 0.37$. $\mu _{\bar{X}}=75,\; \sigma _{\bar{X}}=1.09$. Unless otherwise stated, when we refer to random samples, we assume they are simple random samples. In a random sample, every person in the population has the same chance of being selected. \lim_{n \rightarrow \infty} P(|\overline{X}-\mu| \geq \epsilon)=0. Have questions or comments? Problem 1 : A person deposits $5,000 in a bank account which pays 6% simple interest per year. A high-speed packing machine can be set to deliver between $11$ and $13$ ounces of a liquid. At a signi cance level of = 0:05, is there enough evidence to conclude that the treatment is independent of the side e ect of nausea? This gives you the sample variance. $X_1$, $X_2$, $X_3$, $...$, $X_n$ are independent random variables, and, they have the same distribution, i.e, $$F_{X_1}(x)=F_{X_2}(x)=...=F_{X_n}(x), \qquad \textrm{ for all }x \in \mathbb{R}.$$. You may assume that the normal distribution applies. The proportion of a population with a characteristic of interest is $p = 0.82$. Find the indicated probabilities. This is strong evidence that currently a smaller proportion than $22\%$ smoke.

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