The figure below shows the total number $P(t)$ of Covid-19 cases (in millions) in California on or before day $t$, where $t = 0$ is April 1, 2020. Estimate the $t$-values of three inflection points and explain their significance in terms of the number of daily new cases, approximated by $P'(t)$. There are inflection points at: $t = 120$ $t = 150$ $t = 190$ $t = 230$ $t = 275$ $t = 295$ In terms of the of the number of daily new cases, approximated by $P'(t)$, these correspond to points where the number of new daily cases is zero. peaks or minimums in the number of new daily cases. a change in concavity in the number of new daily cases.
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An inflection point on a curve is a point where the curve changes concavity, i.e., from being concave up (cup-shaped) to concave down (cap-shaped), or vice versa. Show more…
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In the spring of $2003,$ SARS (Severe Acute Respiratory Syndrome) spread rapidly in several Asian countries and Canada. Table 4.9 gives the total number, $P$, of SARS cases reported in Hong Kong $^{17}$ by day $t,$ where $t=0$ is March 17,2003. (a) Find the average rate of change of $P$ for each interval in Table 4.9 (b) In early April $2003,$ there was fear that the disease would spread at an ever-increasing rate for a long time. What is the earliest date by which epidemiologists had evidence to indicate that the rate of new cases had begun to slow? (c) Explain why an exponential model for $P$ is not appropriate. (d) It turns out that a logistic model fits the data well. Estimate the value of $t$ at the inflection point. What limiting value of $P$ does this point predict? (e) The best-fitting logistic function for this data turns out to be $$P=\frac{1760}{1+17.53 e^{-0.1408 t}}$$ What limiting value of $P$ does this function predict? Total number of SARS cases in Hong Kong by day $t$ (where $t=0$ is March 17,2003) $$\begin{array}{c|c|c|c|c|c|c|c}t & P & t & P & t & P & t & P \\\hline 0 & 95 & 26 & 1108 & 54 & 1674 & 75 & 1739 \\5 & 222 & 33 & 1358 & 61 & 1710 & 81 & 1750 \\12 & 470 & 40 & 1527 & 68 & 1724 & 87 & 1755 \\19 & 800 & 47 & 1621 & & & & \\\hline\end{array}$$
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David Hughes would like to determine whether there are more units produced on the afternoon shift than on the morning shift. A sample of 65 morning-shift workers showed that the mean number of units produced was 330. A sample of 67 afternoon-shift workers showed that the mean number of units produced was 327. Assume that the population standard deviation for the number of units produced on the morning shift is 25 and on the afternoon shift is 32. At the 0.06 significance level, is the number of units produced on the afternoon shift larger? Use the tables to determine the z-values (or t-values). Round all z-values to 2 decimal places and t-values to 3 decimal places. Unless otherwise stated, report proportions and probabilities as decimal values (not %) and round them to 4 decimal places. Do not round intermediate results; if you do, round them to 5 decimal places. Step 1. State the null hypothesis and the alternate hypothesis. Step 2. Use α = 0.06. Step 3 & 4. Identify the critical value and formulate the decision rule. Step 5. Make a decision. a. The value of the test statistic is ____. b. The p-value is equal to ____. c. Based on that, your decision is to ____ H0. d. So, your conclusion is that there is ____ that the number of units produced on the afternoon shift is larger.
Jon S.
What is the minimal sample size needed for a 95% confidence interval to have a maximal margin of error of 0.1 in the following scenarios? (Round your answers up the nearest whole number.) (a) a preliminary estimate for p is 0.16 (b) there is no preliminary estimate for p For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. In a random sample of 68 professional actors, it was found that 41 were extroverts. (a) Let p represent the proportion of all actors who are extroverts. Find a point estimate for p. (Round your answer to four decimal places.) (b) Find a 95% confidence interval for p. (Round your answers to two decimal places.) lower limit upper limit Give a brief interpretation of the meaning of the confidence interval you have found. We are 95% confident that the true proportion of actors who are extroverts falls within this interval. We are 5% confident that the true proportion of actors who are extroverts falls above this interval. We are 5% confident that the true proportion of actors who are extroverts falls within this interval. We are 95% confident that the true proportion of actors who are extroverts falls outside this interval. (c) Do you think the conditions np > 5 and nq > 5 are satisfied in this problem? Explain why this would be an important consideration. Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal. No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial. Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial. No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.
Danielle F.
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