confidence intervals for the mean

7/30/2019 Confidence Intervals for the Mean

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Variability: range, inter-quartile range, variance, and standard deviation

The range of a set of data is the difference between the highest and lowest values in the set. The range tells you how spread the entire data is. Problem: Cheryl took 7 math tests in one marking period. What is the range of her test scores?

89, 73, 84, 91, 87, 77, 94

Solution: Ordering the test scores from least to greatest, we get: 73, 77, 84, 87, 89, 91, 94 highest - lowest = 94 - 73 = 21


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Regressi

Suatu kajian di kolej dibuat untuk menjangka C.P.A pelajar tahun I dengan menggunakan gred

pencapaian matematik di sekolah menengah. Berikut adalah data yang diperolehi:

X = 2200 Y = 800

X2

= 250,000, Y2

= 30,000

XY = 90,000 N = 20 = 110, = 27


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Confidence Intervals for the mean (known or n 30)

A point estimate is a specific numerical value estimate of a parameter. The best point estimate of the

population mean is the sample mean xbar.

Three properties of a Good Estimator

1. should be an unbiased estimator.2. should be consistent.3. should be a relatively efficient estimator.

An intervalestimate of a parameter is an interval or a range of values used to estimate the parameter.

This estimate may or may not contain the value of parameter being estimated.

The confidencelevel of an interval estimate of a parameter is the probability that the interval estimate

will contain the parameter.

A confidence interval is a specific interval estimate of a parameter determined by using data obtained

from a sample and by using the specific confidence level of the estimate.

Formula for the Confidence Interval of the Mean for a specific

xbar z/2 (/n) < < xbar + z/2 (/n)


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Example : The dean of a faculty wishes to estimate the average age of the students enrolled. From past

studies, the standard deviation is known to be 2 years. A sample of 50 students is selected, and the

mean is found to be 23.2 years. Find the 95% confidence interval of the population mean.

Contoh: dekan fakulti ingin untuk menganggarkan umur purata pelajar mendaftar. Dari kajian yang lalu, sisihan piawai dikenali untuk menjadi 2 tahun. Satu

sampel 50 orang pelajar dipilih, dan min didapati menjadi 23,2 tahun. Cari selang keyakinan 95% penduduk bermakna.

Hence, the dean can say, with 95% confidence, that the average age of students is between 22.6 and

23.8 years, based on 50 students.

Oleh itu, Dekan boleh katakan, dengan keyakinan 95%, bahawa purata umur pelajar adalah antara 22,6 dan 23.8 tahun, berdasarkan 50 orang pelajar.

Example : A certain medication is known to increase the pulse rate of its users. The standard deviation

of the pulse rate is known to be 5 beats per minutes. A sample of 30 users had an average pulse rate of

104 beats per minutes. Find the 99 % confidence interval of the true mean.

Contoh: Satu ubat tertentu dikenali untuk meningkatkan kadar nadi pengguna. Sisihan piawai kadar nadi dikenali sebagai 5 denyutan per minit. Satu sampel

sebanyak 30 pengguna mempunyai kadar nadi purata 104 degupan per minit. Dapatkan selang keyakinan 99% min sebenar.

Hence, one can be 99% confidence that the mean pulse rate of all users of the medication is between

102 and 106 beats per minutes, based on a sample of 30 users.

Justeru, seseorang itu boleh menjadi keyakinan 99% bahawa purata kadar nadi semua pengguna ubat adalah antara 102 dan 106 degupan per minit,

berdasarkan sampel sebanyak 30 pengguna.


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Example :

Hence, one can be 90% confident that the population mean of the assets of all the credit unions is

between $6.752 million and 15.430 million, based on a sample 30 credit unions.

Justeru, seseorang itu boleh menjadi 90% yakin bahawa min populasi aset semua kesatuan kredit adalah antara $ 6752000 dan 15,430 juta, berasaskan pada

kesatuan sampel kredit 30.


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Formula to the Minimum Sample Size Needed for an Interval Estimate of the Population Mean

Example :The Faculty Dean asks the statistic teacher to estimate the average age of the students at their

faculty. How large sample is necessary? The statistic teacher would like to be 99% confident that the

estimate should be accurate within one year. From previous study, the standard deviation of ages is

known to be 3 years.

Contoh: Dekan Fakulti meminta guru statistik untuk menganggar purata umur pelajar di fakulti mereka. Bagaimana sampel besar adalah perlu? Guru statistik

ingin menjadi 99% yakin bahawa anggaran harus tepat dalam tempoh satu tahun. Daripada kajian sebelumnya, sisihan piawai umur dikenali sebagai 3 tahun.


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Confidence Intervals for the Mean ( Unknown and n< 30)

Find the t/2 value for a 95% confidence interval when the sample size is 22

Xbar t/2 (s/n)


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Example: Ten randomly selected automobiles were stopped, and the tread depth of the right front tire

was measured. The mean was 0.32 inch, and the standard deviation was 0.08 inch. Find the 95%

confidence interval of the mean depth. Assume that the variable is approximately normally distributed.

Sepuluh kereta yang dipilih secara rawak telah dihentikan, dan kedalaman bunga tayar depan kanan diukur. Min adalah 0,32 inci, dan sisihan piawai adalah

0,08 inci. Cari selang keyakinan 95% kedalaman min. Andaikan bahawa pembolehubah hampir bertabur normal.

Therefore, one can be 95% confident that the population mean tread depth of all right front tires is

between 0.26 and 0.38 inch based on 10 tires

Oleh itu, seseorang boleh menjadi 95% yakin bahawa min populasi kedalaman bunga semua tayar depan kanan adalah antara 0,26 dan 0,38 inci

berdasarkan pada 10 tayar


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Example: The data represent a sample of the number of home fires started by candles for the past

several years. (Data are from the National Fire Protection Association). Find the 99% confidence interval

for the mean number of home fires started by candles each year.

5460 5900 6090 6310 7160 8440 9930

Data yang mewakili sampel bilangan kebakaran rumah yang bermula dengan lilin untuk beberapa tahun yang lalu. (Data dari Persatuan Perlindungan

Kebakaran Kebangsaan). Dapatkan selang keyakinan 99% bagi min bilangan kebakaran rumah bermula dengan lilin setiap tahun.

One can be 99% confident that the population mean of home fires started by candles each year is

between 4785.2 and 9297.6, based on sample of home fires occurring over a period of 7 years.

seseorang boleh menjadi 99% yakin bahawa min populasi kebakaran rumah yang dimulakan oleh lilin setiap tahun adalah antara 4785,2 dan 9297,6,

berdasarkan sampel kebakaran rumah berlaku sepanjang tempoh 7 tahun.


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Confidence Intervals for Variances and Standard Deviation

Example: Find the 95% confidence interval for the variance and standard deviation of the nicotine

content of cigarettes manufactured if a sample of 20 cigarettes has a standard deviation of 1.6

miligrams.

Cari selang keyakinan 95% bagi varians dan sisihan piawai kandungan nikotin rokok dibuat jika sampel sebanyak 20 rokok mempunyai sisihan piawai

sebanyak 1.6 miligrams.

Hence, one can be 95% confident that the true variance for the nicotine content is between 1.5 and 5.5

milligrams.

Justeru, seseorang itu boleh menjadi 95% yakin bahawa varians sebenar untuk kandungan nikotin adalah antara 1.5 dan 5.5 miligram.

Hence, one can be 95% confident that the true standard deviation for the nicotine content of all

cigarettes manufactured is between 1.2 and 2.3 milligrams based on a sample of 20 cigarettes.

Justeru, seseorang itu boleh menjadi 95% yakin bahawa sisihan piawai bagi kandungan nikotin semua rokok yang dihasilkan adalah di antara 1.2 dan 2.3

miligram berdasarkan sampel sebanyak 20 rokok.


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Example:

Find the 90% confidence interval for the variance and standard deviation for the price of an adult single-

day ski ticket. The data represent a selected sample of nationwide ski resorts. Assume that the variable

is normally distributed.

$59 54 53 52 51

$39 49 46 49 48

Dapatkan selang keyakinan 90% bagi varians dan sisihan piawai bagi harga tiket ski dewasa dalam satu hari. Data yang mewakili sampel yang dipilih resort ski

di seluruh negara. Menganggap bahawa pembolehubah biasanya diedarkan.

Hence, one can be 90% confident that the true variance for the price of all single day ski lift tickets of the

population is between $15 and $76.30 based on a sample of 10 nationwide ski resort.

Justeru, seseorang itu boleh menjadi 90% yakin bahawa varians sebenar bagi harga tiket lif ski sepanjang hari tunggal daripada penduduk adalah di antara $

15 dan $ 76,30 berdasarkan sampel 10 ski resort di seluruh negara.

Hence, one can be 90% confident that the standard deviation for the price of all single day ski lift tickets

of the population is between $3.87 and $8.73 based on a sample of 10 nationwide ski resort.

Justeru, seseorang itu boleh menjadi 90% yakin bahawa sisihan piawai untuk harga tiket lif ski sepanjang hari tunggal daripada penduduk adalah antara $

3,87 dan $ 8,73 berdasarkan sampel 10 ski resort di seluruh negara.


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confidence intervals for the mean

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