A state public health department wishes to investigate the effectiveness of a campaign against smoking. You may assume that the normal distribution applies. First verify that the sample is sufficiently large to use the normal distribution. The mean for a sample is derived using Formula 3.4. You may assume that the normal distribution applies. f_X(x)=1, \qquad \textrm{ for }x \in [0,1], Compare this with the variance and population standard deviation for the same data. This gives you the sample variance. Find the mean and standard deviation of the sample proportion \(\widehat{P}\), The proportion of a population with a characteristic of interest is \(p = 0.37\). A population has mean \(128\) and standard deviation \(22\). \end{align} When a biologist wishes to estimate the mean time that such sharks stay immobile by inducing tonic immobility in each of a sample of \(12\) sharks, find the probability that mean time of immobility in the sample will be between \(10\) and \(13\) minutes. A population has mean \(5.75\) and standard deviation \(1.02\). ;�= How strong is the evidence that the campaign to reduce smoking has been effective? If the mean is so low, is that particularly strong evidence that the tire is not as good as claimed? Random samples of size \(64\) are drawn from a population with mean \(32\) and standard deviation \(5\). %PDF-1.5 \end{align} Nevertheless, for the material that we cover in this book simple random sampling is sufficient. Find the probability that the mean of a sample of size \(9\) drawn from this population exceeds \(30\). Example 1: A school has 650 students. These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang. Random samples of size \(1,600\) are drawn from a population in which the proportion with the characteristic of interest is \(0.05\). (4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9) / 19 = 178/19 = 9.368, The population standard deviation is the square root of the variance. Find the probability that in a random sample of \(250\) men at least \(10\%\) will suffer some form of color blindness. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Simple random sampling: By using the random number generator technique, the researcher draws a sample from the population called simple random sampling. Find the mean and standard deviation of the sample proportion \(\widehat{P}\), The proportion of a population with a characteristic of interest is \(p = 0.76\). Suppose that \(29\%\) of all residents of a community favor annexation by a nearby municipality. We chose a person uniformly at random from the population and let $X_1$ be the height of that person. You may assume that the normal distribution applies. The big advantage of sampling with replacement (the above procedure) is that $X_i$'s will be independent and this makes the analysis much simpler. \(\mu _{\bar{X}}=75,\; \sigma _{\bar{X}}=1.09\). He knows that five years ago, \(38\%\) of all passenger vehicles in operation were at least ten years old. 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. Practice Problem 2: The side e ects of a new drug are being tested against a placebo. Find the probability that the mean of a sample of size \(30\) will be less than \(72\). In an effort to reduce the population of unwanted cats and dogs, a group of veterinarians set up a low-cost spay/neuter clinic. Central Limit Theorem: The random variable Zn = ¯ X − μ σ / √n = X1 + X2 +... + Xn − nμ √nσ converges in distribution to the standard normal random variable as n goes to infinity, that is lim n → ∞P(Zn ≤ x) = Φ(x), for all x ∈ R where Φ(x) is the standard normal CDF. We have 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. Find the probability that average lifetime of the five tires will be \(57,000\) miles or less. \(\mu _{\widehat{P}}=0.37,\; \sigma _{\widehat{P}}=0.012\), \(\mu _{\widehat{P}}=0.76,\; \sigma _{\widehat{P}}=0.012\), \(p\pm 3\sqrt{\frac{pq}{n}}=0.25\pm 0.087,\; \text{yes}\), \(\hat{p}\pm 3\sqrt{\frac{\hat{p}\hat{q}}{n}}=0.48\pm 0.21,\; \text{yes}\), \(\hat{p}\pm 3\sqrt{\frac{\hat{p}\hat{q}}{n}}=0.12\pm 0.14,\; \text{no}\), \(\hat{p}\pm 3\sqrt{\frac{\hat{p}\hat{q}}{n}}=0.12\pm 0.10,\; \text{yes}\), \(p\pm 3\sqrt{\frac{pq}{n}}=0.08\pm 0.05\) and \([0.03,0.13]\subset [0,1],0.1210\), \(p\pm 3\sqrt{\frac{pq}{n}}=0.02\pm 0.01\) and \([0.01,0.03]\subset [0,1],0.9671\). Give an interpretation of the result in part (b). F_X(x)=x, \qquad \textrm{ for }x \in [0,1]. Suppose this proportion is valid for all homes. Suppose speeds of vehicles on a particular stretch of roadway are normally distributed with mean \(36.6\) mph and standard deviation \(1.7\) mph. Let $X_{(1)}, X_{(2)}, \cdots, X_{(n)}$ be the order statistics of $X_1$, $X_2$, $X_3$, $...$, $X_n$. \begin{align}%\label{} When collecting data, we often make several observations on a random variable. Suppose that in a particular species of sharks the time a shark remains in a state of tonic immobility when inverted is normally distributed with mean \(11.2\) minutes and standard deviation \(1.1\) minutes.

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