Statistical Applets

You can use the sliders to change both *n* and *p*. Click and drag a slider with the mouse. Start by choosing *p*. The binomial distributions are symmetric for *p* = 0.5. They become more skewed as *p* moves away from 0.5. The bars show the binomial probabilities. The vertical gray line marks the mean *np*. The red curve is the normal density curve with the same mean and standard deviation as the binomial distribution. As you increase *n*, the binomial probability histogram looks more and more like the normal curve.

Click the "Quiz Me" button to complete the activity.

The Central Limit Theorem says that as *n* increases, the binomial distribution with *n* trials and probability *p* of success gets closer and closer to a normal distribution. That is, the binomial probability of any event gets closer and closer to the normal probability of the same event. The normal distribution has the same mean *μ = np* and standard deviation as the binomial distribution.

Suppose you flip a "fair" coin (that is, one with probability 0.5 of coming up either heads or tails) 16 times. According to the applet, the most likely result will be that L6bSXEGJIC8= of the tosses will come up heads. The probability of this outcome, according to the applet, is approximately eVl1o4RLWZscZlBuQFUlkYJjl4TkN8LtG1MQOqnfHmc=.

2

Try again.

Incorrect. See above for the correct answers.

Great job.

Suppose you flip a "fair" coin (that is, one with probability 0.5 of coming up either heads or tails) 16 times. According to the applet, the most likely result will be that ____ of the tosses will come up heads. The probability of this outcome, according to the applet, is approximately ____.

Now increase the number of trials to 50. According to the applet, the most likely result will be that Bqf/EtUReUc= of the tosses will come up heads. Compared to the distribution of results with 16 trials, the distribution with 50 trials j1EopTmnsJsYHFr1WikKhp5fyU2aEd59JcIqxTJvzQI= resembles the normal curve, because with more possible outcomes, the distribution is L4UsIC4XF2D9vWacd/4VPvEEFquRFD1v.

2

Try again.

Incorrect. See above for the correct answers.

Great job.

Now increase the number of trials to 50. According to the applet, the most likely result will be that ____ of the tosses will come up heads. Compared to the distribution of results with 16 trials, the distribution with 50 trials ____ resembles the normal curve, because with more possible outcomes, the distribution is ____.

The most likely outcome for any given probability *p* and number of trials *n* is *p* times *n*, so the most likely outcomes are that you will toss 2, 5, and 7 heads if the probability of heads is 0.2, 0.5, or 0.7 respectively.