# Distribution

Classified in Mathematics

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Sample mean: x bar

Descriptive- organizing,summarizing, presenting data

Inferential – making conclusions and inferences about The data

scores | Frequency | Relative frequency | Cumaltive relative frequency |

3 | 3/15 | 0.2 | |

4 | 4/15 | 0.4667 | |

5 | 5/15 | 0.4667+0.333=0.8 | |

2 | 2/15 | 0.9333 | |

1 | 1/15 | 1 |

Box plot: q1 – mean –q3

**A fitness
Center is interested in the mean amount of time a client exercises in the
Center each week. **

**population**: all clients in fitness center

**sample**: group of these clients

parameter: population mean amount of **time** clients
Exercise in center each week

statistic: sample mean amount of time clients exercise In center each week

variable: X=the amount of time one client exercises in The center each week

data: values for X (4 hours, 6 hours, 10 hours, etc.)

**number of
Tickets sold to a concert **Quantitative discrete, 100 **percent of body fat **Quantitative continuous, 10% ** favorite baseball team **

Qualitative, Dodgers

The t **distribution** is flatter in the middle and the
Tails are wider.

The more degrees of freedom we have, the closer we get To the actual number.

The difference in the hypothesis test is that the Critical region changes and the normal curve uses +/- 1.96.

Increase the variability of the scores :

Increases denominator and t is closer to zero

Increase the number of scores in the sample :

Decreases the Sm in the denominator, and increases t.

Increase the difference between the sample mean and the Population mean:

Increases numerator and increases t.

Sample **standard** deviation is the average distance
Of all of the data from the mean in the population. The standard error is the
Average distance of all the samples of the size of the data (n) in the
Population mean.

5. - Ho µ phys. Ed test=12

H1 µ phys. Ed test≠12

Two tailed, a=.05

Df =25-1= 24 Critical region: ±2.064

T = 15-12/29/√25= 1.67

T(24)=1.67, p>.05

We fail to reject the null hypothesis Because there is no significant effect of the P.E programs on pushup scores, on Average, students who are in P.E programs do not perform better. M=15, SD=1.67

The heights of
The 430 National Basketball Association players were listed on team rosters at
The start of the 2005–2006 season. The heights of basketball players have an
Approximate normal distribution with mean µ = 79 inches and a standard
Deviation σ = 3.89 inches. For each of the following heights, calculate the
Z-**score** and interpret it using complete sentences.

77 inches z= 77-79/3.89 in. = -2/3.89= -2/3.89 = -0.5141The height of 77 in. Is 0.5141 standard deviations Below the mean. 77 inches is shorter than average.

The 90^{th} percentile for recovery times is?7.99-5.3/2.1=2.69+2.1=1.2801 à 0.39973+.5= 0.89973 à .90

In national use, a vocabulary test is Known to have a mean score of 68 and a standard deviation of 13. A class of 19 Students takes the test and has a mean score of 65. What is the probability That the 19 students will obtain the mean of 65 or higher?

Z=65-68/13/√19=-3/2.98240=-1.01 à.34375+.5=

.84375

For a sample of n = 25 scores, what is the Probability that the sample mean will be within 5 points of the population Mean? In other words, what is p(95 < < 105)?

Z = (M - μ) / √(σ2 / n)

Z For 95 à95-100/225/√25 =-5/3=-1.66667

Z for 100 à 105-100/20/√25 = 5/3 =1.66667

Law of large numbers - if You take samples of larger and larger size from any population, then the mean x ¯ of the sample tends to get closer and closer to μ

Sampling distribution- if You draw random samples of size n, the distribution of the sample means, that Is called the sampling distribution of the mean. The sampling distribution of The mean approaches a normal distribution as n, the sample size, increases.

Central limit theorem- the Larger the sample size, the more normal the distribution will be.

Standard error-the standard Deviation of the distribution of the sample means

formulas: z=x-u/o z=x-u/o/sq root of n

t=x-u/s/sq root two sample independent t: t=(x1-x2)-(u1-u2)/s(x1-x2)

t=x1-x2/sq root(n1-1)s1squared+(n2-1)s2squared/n1+n2-2(n1+n2)/n1n2

formula for dependent measures: t=ed/sq root nED squared-(ED)squared/n-1

Houcomf=u not comf

h1=ucomf=not comf

t=x1-x2 ... =-3.27

reject null, look at means shorter height more comfy

- 1 sample measured two times depends on T more control and size of group (advantages of dependent t test)

we use chi square or nonparametric stats when data comes as frequencies, it is measured at nominal or categorical level

prego not prego total -

clown 33 60 93 93(51)/186=25.5 --- 33-25.5squared/35.5=2.21 add and compare

no clown 18 75 93

total : 51 135 186