Aplikasi Sebaran Normal Pertemuan 12 Matakuliah: L0104 / Statistika Psikologi Tahun : 2008

Bina Nusantara Learning Outcomes 3 Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa akan dapat menghitung peluang Binomial dengan pendekatan pada sebaran normal baku.

Bina Nusantara Outline Materi 4 Metode deskriptif untuk sebaran normal pendekatan sebaran Binomial pada sebaran normal baku Koreksi kekontinuan

Bina Nusantara The Normal Approximation to the Binomial We can calculate binomial probabilities using –The binomial formula –The cumulative binomial tables –Do It Yourself! applets When n is large, and p is not too close to zero or one, areas under the normal curve with mean np and variance npq can be used to approximate binomial probabilities.

Bina Nusantara Approximating the Binomial continuity correction. Make sure to include the entire rectangle for the values of x in the interval of interest. This is called the continuity correction. Standardize the values of x using Make sure that np and nq are both greater than 5 to avoid inaccurate approximations!

Bina Nusantara Example Suppose x is a binomial random variable with n = 30 and p =.4. Using the normal approximation to find P(x  10). n = 30 p =.4 q =.6 np = 12nq = 18 The normal approximation is ok!

Bina Nusantara Example Applet

Bina Nusantara Example A production line produces AA batteries with a reliability rate of 95%. A sample of n = 200 batteries is selected. Find the probability that at least 195 of the batteries work. Success = working battery n = 200 p =.95 np = 190nq = 10 The normal approximation is ok!

Bina Nusantara Central Limit Theorem: If random samples of n observations are drawn from a nonnormal population with finite  and standard deviation , then, when n is large, the sampling distribution of the sample mean is approximately normally distributed, with mean  and standard deviation. The approximation becomes more accurate as n becomes large. Central Limit Theorem: If random samples of n observations are drawn from a nonnormal population with finite  and standard deviation , then, when n is large, the sampling distribution of the sample mean is approximately normally distributed, with mean  and standard deviation. The approximation becomes more accurate as n becomes large. Sampling Distributions Sampling distributions for statistics can be Approximated with simulation techniques Derived using mathematical theorems The Central Limit Theorem is one such theorem.

Bina Nusantara Example uniform. Toss a fair coin n = 1 time. The distribution of x the number on the upper face is flat or uniform. Applet

Bina Nusantara Example mound- shaped. Toss a fair coin n = 2 time. The distribution of x the average number on the two upper faces is mound- shaped. Applet

Bina Nusantara Example approximately normal. Toss a fair coin n = 3 time. The distribution of x the average number on the two upper faces is approximately normal. Applet

Bina Nusantara Why is this Important? Central Limit Theorem The Central Limit Theorem also implies that the sum of n measurements is approximately normal with mean n  and standard deviation. Many statistics that are used for statistical inference are sums or averages of sample measurements. normal When n is large, these statistics will have approximately normal distributions. reliability This will allow us to describe their behavior and evaluate the reliability of our inferences.

Bina Nusantara How Large is Large? normal If the sample is normal, then the sampling distribution of will also be normal, no matter what the sample size. symmetric When the sample population is approximately symmetric, the distribution becomes approximately normal for relatively small values of n. (ex. n=3 in dice example) skewed at least 30 When the sample population is skewed, the sample size must be at least 30 before the sampling distribution of becomes approximately normal.

Bina Nusantara The Sampling Distribution of the Sample Proportion Central Limit Theorem The Central Limit Theorem can be used to conclude that the binomial random variable x is approximately normal when n is large, with mean np and standard deviation. The sample proportion, is simply a rescaling of the binomial random variable x, dividing it by n. approximately normal, From the Central Limit Theorem, the sampling distribution of will also be approximately normal, with a rescaled mean and standard deviation.

Bina Nusantara The Sampling Distribution of the Sample Proportion The standard deviation of p-hat is sometimes called the STANDARD ERROR (SE) of p-hat. A random sample of size n is selected from a binomial population with parameter p   he sampling distribution of the sample proportion, will have mean p and standard deviation approximately normal. If n is large, and p is not too close to zero or one, the sampling distribution of will be approximately normal.

Bina Nusantara Finding Probabilities for the Sample Proportion If the sampling distribution of is normal or approximately normal  standardize or rescale the interval of interest in terms of Find the appropriate area using Table 3. If the sampling distribution of is normal or approximately normal  standardize or rescale the interval of interest in terms of Find the appropriate area using Table 3. Example: Example: A random sample of size n = 100 from a binomial population with p =.4.

Bina Nusantara Example The soda bottler in the previous example claims that only 5% of the soda cans are underfilled. A quality control technician randomly samples 200 cans of soda. What is the probability that more than 10% of the cans are underfilled? This would be very unusual, if indeed p =.05! n = 200 S: underfilled can p = P(S) =.05 q =.95 np = 10 nq = 190 n = 200 S: underfilled can p = P(S) =.05 q =.95 np = 10 nq = 190 OK to use the normal approximation

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