# A random t-Distribution problem

(10 points)

Use this webpage to generate some personal random data. Then,

- Write down a 90% confidence interval for your data and
- Test the null hypothesis that the mean of your data is 50 against the alternative hypothesis that the mean of your data is greater than 50 - again at the 90% level. Be sure to

(a) Compute the mean, standard deviation, standard error, test statistic, and p-value for your test, and

(b) Draw the appropriate conclusion stating the reason for your conclusion

You can use our mean and standard deviation calculator as well as our t-Distribution calculator.

## Comments

My data is below:

53.88 51.15 54.97 51.43 53.34

58.92 55.26 53.81 53.56 53.66

48.69 57.60 52.11 49.78 58.07

My Standard deviation for this data is 2.93271660183296

My mean for this sample is 53.74866666666666

For a 90% confidence interval, my t* multiplier is 1.7291327925089004

My $ME= 1.72913times(2.9327/sqrt15)=1.3093$

My confidence interval is then (53.74866-1.3093, 53.74866+1.3093) = (52.43936, 55.05796)

The setup for my hypothesis test is below:

$H_0: u=50$

$H_A: u>50$

My standard error = $2.3927/sqrt15= .61779$

My test statistic is $(53.74866-50)/(2.9327/sqrt15) = 4.9505$

My p-value for the above test statistic is 0.9998933, but I want the P-value for u>50, so I do (1-that p-value) to get 0.00010662456827988486.

That value is smaller than the .1 indicated from our confidence interval, so I reject the null hypothesis.

My data is below:

51.78 54.62 54.09 53.15 55.96

55.23 53.15 55.26 55.48 51.23

53.53 57.25 52.73 53.48 59.63

The standard deviation of the data is 2.155677553412318

The mean of the sample is 54.438

With a 90% confidence interval, the t* multiplier is 1.7613101151015698

$ME =1.76131times(2.1556/sqrt15)=.9802$

The confidence interval is (54.438-.9802, 54.438+.9802) = (53.4578, 55.4182)

$H_0: U=50$

$H_A: U>50$

Standard Error: $2.1556/sqrt15= .55657$

Test Statistic: $(54.438-50)/(2.1556/sqrt15) = 7.9737$

The p-Value for the test statistic is 0.9999992, but I want the p-Value for u>50, so I subtract that value from 1 and arrive at 0.00010662456827988486.

This value is smaller than the .1 indicated from the confidence interval, so I would reject the null hypothesis.

My data is below:

52.13 53.26 48.12 52.55 56.23

52.99 53.12 49.94 62.93 56.69

49.82 52.53 56.21 56.09 52.66

My standard deviation for this data is 3.5871095938227535

My mean for this sample is 53.684666666666665

For a 90% confidence interval, y t* multiplier is 1.7613101151015698

My ME= 1.7613 x (3.587/√15) = 1.631

My confidence interval is then (53.6846-1.631, 53.6946+1.631) = (52.053, 55.316)

The setup for my hypothesis test is below:

H0:u=50

HA:u>50

My standard error 3.587/√15= 0.9262

My test statistic is 53.6846-50/3.587/√15= 3.5689

My p-value for the above test statistic is 0.998458 but I want the P-value for u>50, so I do (1-that p-value) to get 0.00010662456827988486

That value is smaller than the .1 indicated from our confidence interval

I reject the null hypothesis.

My data is below

58.24 52.97 51.93 50.31 51.92

53.10 50.59 56.89 57.21 54.17

54.80 45.68 56.56 60.74 53.82

Standard deviation: 3.7243290223430403

Mean: 53.928666666666665

T*= 1.7613101151015698

My ME is: 1.7613 x ( 3.724/ sqrt15 ) = 1.6935

Confidence interval is: (53.9286-1.6935,53.9286+1.6935)= (52.2351,55.6221)

H0: u=50

HA: u>50

Standard error 3.724/ sqrt15 = .9615

Test statistic: (53.9286-50)/(3.724/ sqrt15) = 4.0858

my P value is .9995130488756515

1-.9995130488756515 = .0005

I reject my null hypothesis

My data is below:

55.14 57.88 47.90 55.23 48.82

46.80 56.05 50.73 57.71 52.63

56.44 53.17 51.47 55.93 55.03

My mean for my data set is 53.39533333333334

My standard deviation is 3.551966993626141

For a 90% confidence interval, the t* multiplier is 0.9984057759717236

My margin of error would be found as follows:

$ME=0.99841(3.55196/sqrt15)=0.9156539201$

My confidence interval would then be:

$(53.39533-0.91565, 53.39533+0.91565) = (52.47968, 54.31098)$

My hypothesis would then be set up as follows:

$H0:u=50$

$Ha:u>50$

My standard error is $(3.55196/sqrt15) = 0.9173703273$

My test statistic is $(53.3953-50)/(3.55296/sqrt15) = 3.701122544$

My p-value for the above statistic is 0.9988139785333616, but I wanted P-value for $ha:u>50$ so I subtract my p-value from 1 to get 0.001186021467

That value is smaller than .9 indicated by our confidence interval, so I reject the null hypothesis.

I was given the following data:

58.29 53.64 54.04 47.38 60.60

51.96 47.41 53.45 49.35 54.57

45.69 52.53 56.72 53.63 51.83

The mean is 52.73933333333333

The standard deviation of the data is 4.100359682248562

My t* multiplier is 1.7613101151015698 with a 90% confidence interval

ME =1.76131

(4.1004/sqrt(15))=1.8641.864),52.73933+(1.7613*1.864)][xˉ−t ∗ ME, xˉ +t ∗ ME],

[52.73933-(1.7613

[49.456,56.022]

My hypothesis is set up as follows:

H0:u=50

Ha:u>50

My standard error is (4.1004/sqrt(15))=1.058

My test statistic is (52.7393−50)/(1.058)=2.589

My initial p-value for this data is 0.9892844500302621, however after subtracting this from 1, I get my true p-value of 0.0108

I reject the null hypothesis because my given value is smaller than 0.1

My data was:

50.02 54.21 54.74 53.33 55.28

57.20 52.15 54.70 58.64 50.89

51.49 51.02 60.44 48.99 54.27

Mean: 53.824666666666666

Std Dev: 3.220909070137689

My t* was 1.761310

My Margin of Error = 1.761310 x(3.2209/√15)=1.46476

My Confidence interval is: (53.8247+1.46476,53.8247-1.46476)= (52.36,55.289)

The setup for my hypothesis test is below:

H0:u=50

HA:u>50

My standard error = 3.22/√15= .8314

My test statistic = 53.8247-50/3.22/√15=4.981

My p-value for the above test statistic is 0.9998626, but I want the P-value for u>50, so I do (1-that p-value) to get 0.00010662456827988486.

i reject my null hypothesis

My data is below:

53.72 54.22 51.82 54.77 55.09

52.21 52.27 49.98 51.40 52.50

54.03 58.26 52.57 54.14 54.65

My standard deviation for this data is 1.9638272

My mean for this sample is 53.442

For a 90% confidence interval,t* multiplier is 1.7291327925089004

My ME=1.72913×(1.9638272−−√15)= .876769

My confidence interval is then (53.442-.876769,53.442+.876769)=(52.565231, 54.318769

The setup for my hypothesis test is below:

H0:u=50

HA:u>50

My standard error = 1.9638272/√15= .5070580027

My test statistic is 53.442-50/1.9638272/√15=6.788178

My p-value for the above test statistic is 0.9998626, but I want the P-value for u>50, so I do (1-that p-value) to get 0.00010662456827988486.

That value is smaller than the .1 indicated from our confidence interval, so I reject the null hypothesis.

My data is:

57.74 58.29 53.28 49.36 52.74

58.15 55.05 58.04 48.90 55.37

56.26 53.22 52.93 51.35 54.82

Mean: 54.36666666666667

SD: 3.0601112647060646

t∗ =1.7613101151015698

ME = 1.76131 x (3.0601/√15) = 1.39

Confidence Interval:

[54.3666 - 1.39, 54.3666 + 1.39] = [52.9766, 55.7566]

Hypothesis Test:

H0: μ = 50

HA: μ > 50

Standard Error:

3.0601/√15 = .7901

Test Statistic:

54.3666-50 / (3.0601/√15) = 5.5267

P-Value:

0.9999627069218578

I want the P-value for u>50, so I calculate 1 - 0.9999627069218578 for a P-Value of 0.00003729307. I reject my null hypothesis.

My random data is:

51.33 51.63 54.35 55.22 53.38

49.87 57.66 51.69 54.39 52.99

54.52 55.93 54.41 60.95 56.93

Mean: 54.35

Standard Deviation: 2.814696127

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

ME= 1.72913times(2.8146/sqrt15)=1.2566

My confidence interval is (54.35-1.2566, 53.0934+1.2566) = (52.43936, 55.6066)

The setup for my hypothesis test is below:

H0:u=50

Ha:u>50

Standard error: 2.8146/sqrt15= .7267265951

Test Statistic: (54.35-50)/(2.8146/sqrt15) = 5.985744886

P-value: 0.9473

1-the P-value: 0.0527

This value is smaller than the .1 indicated from our confidence interval, so I reject the null hypothesis.

My Random data is:

55.13 52.91 54.46 54.25 56.19

46.04 54.57 53.95 52.46 57.37

57.21 52.50 49.38 53.81 50.11

Mean: 53.356

Standard Deviation: 3.021

For a 90% confidence interval, my t* multiplier is 1.761

My margin of error would be found as follows:

ME= 1.761(3.021/√15)= 1.374

My confidence interval would then be:

(53.356−1.374+1.374)=(51.98, 54.73)

Hypothesis Test:

H0: μ = 50

HA: μ > 50

Standard Error: 3.021/√15 =.78

Test Statistic: 53.356−50/(3.021/√15)= 4.302

The p-Value for the test statistic is 0.9996346444691792, but I want the p-Value for u>50, so I subtract that value from 1 and arrive at 0.0003653555308208935.

This value is smaller than then 1 indicated from the confidence interval, so I would reject the null hypothesis.

My random data is:

55.71 49.98 54.34 51.48 49.47

55.03 50.69 49.28 52.30 54.00

51.46 54.79 60.08 56.08 50.84

Standard deviation:3.0196046729584287

Mean:53.03533333333335

For a 90% confidence interval the t* multiplier is: 1.7613101151015698

My margin of error= 1.76131x(3.0196/sqrt15) =1.3732

My confidence interval is: (53.0353-1.3732, 53.0353+1.3732)= (51.6621, 54.4085)

The setup for my hypothesis test is below:

H0: u=50

HA: u>50

My standard error= 3.0196/sqrt15= .7797

My test statistic= (53.0353-50)/(3.0196/sqrt15)=3.8929

My p-value: .9991

1-p-value: 0.00081

This value is smaller than the .1 indicated from our confidence interval, so I reject the null hypothesis.

My data below is:

50.83 49.59 54.24 57.24 54.15

52.24 52.83 56.13 50.71 50.27

51.11 57.58 53.70 57.07 54.94

Mean:=53.50866666666666

STD DEV= 2.696041719332427

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

ME = 1.76131 * (2.69604/sqrt 15)=1.22607

Confidence Interval:

[53.50867 - 1.22607 = 52.2826] [53.50867 + 1.22607= 54.73474]

The setup for my hypothesis test is below:

H0:u=50

HA:u>50

Standard Error:

2.69604/√15 = 0.69611

Test Statistic:

53.50867-50 / (2.69604/√15) = 5.040395914

pvaule:0.99990975

1 - the pvaule: 0.0902475

This value is smaller than the .1 indicated from our confidence interval, so I reject the null hypothesis.

My randomize data is:

55.00 51.57 52.80 54.61 53.26

54.48 49.70 53.41 61.72 51.41

56.04 48.48 55.59 54.66 54.22

Mean: 53.79666666666666

Standard Deviation: 3.0666819357797923

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

ME: $1.7613101151015698(3.067/sqrt15)= 1.3948$

My confidence interval is (53.7967- 1.3948; 53.7967+ 1.3948) = (52.4019, 55.1915)

Hypothesis Test:

H0: u=50

Ha: u>50

Standard Error:

$(3.067/sqrt15)= 0.7919$

Test Statistic:

$(53.7967-50)/(3.067/sqrt15)= 4.794$

p-value:

0.0001428847517788286

This value is smaller than 0.1 indicator so we reject the null hypothesis

My random data is:

55.81 56.66 58.67 51.26 53.55

59.96 56.22 53.59 55.22 57.49

58.54 57.40 48.84 54.65 49.76

mean =55.174666

Std dev= 3.280428688594279

for a 90% confidence interval the t* multiplier is: 1.34503038

My Margin of error: 1.139244828

My confidence interval: 56.31391083, 54.03542117

H0:u=50

HA:u>50

My standard error = .8470030453

My test statistic = 55.174666-50/3.22/sqrrt of 15 (dont know how to type square root)= .4072921995

My p-value for the above test statistic is , but i want the p-value for u>50, so i do 1-that p-value and i get 0.6550236737921495

i reject my null hypothesis

Your random data is:

50.87 55.84 55.88 58.01 52.51

55.58 57.32 53.51 55.48 50.31

53.93 52.81 51.10 57.65 55.01

Mean:54.48

Standard Deviation: 2.49

T*: 1.76

ME: 1.13

My Confidence Interval is $(54.48-1.13, 54.48+1.13)$ = (53.35, 55.61)$

Hypothesis Test:

H0:u=50

H0:u 50

Standard Error: $2.49/Squr15=9.64372853206$

Test Static: (54.48-50)/(2.49/sqrt15)= .464550612878

P-Value: 0.6384

1-P-Value: 0.3616

We Reject the null hypothesis

My random data is:

50.74 54.21 55.62 50.96 53.59

53.23 53.37 57.34 48.97 54.13

54.10 51.83 51.97 52.39 59.49

Mean:53.46266666666668

std de : 2.637776300129299

The t* multiplier is: 1.7613101151015698

ME= (1.72913*2.6377763/sqrt15 ) =1.177660

My confidence interval is: (53.46 - 1.177660) , (53.46 + 1.177660) = (52.28234 , 54.633766)

Test Statistic : (53.46- 50)/(2.6377763/sqrt15) = 5.0802

P value=0.9999

1- P value =0.0000678

My random data is:

58.52 54.18 54.67 57.90 54.73

52.48 53.75 55.29 56.61 56.10

56.93 52.95 49.56 56.21 51.94

Mean: 54.788

Standard Deviation: 2.407254512035295

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

$ME= 1.7613101151015698(2.407254512035295/sqrt15)=1.094743$

My confidence interval is (54.788+1.094743, 54.788-1.094743) = (55.882743, 53.693257)

The setup for my hypothesis test is below:

H0:u=50

Ha:u>50

$SE=2.407254512035295/sqrt15= .6215504423$

$Test Statistic= (54.788-50)/.6215504423 = 7.703316858$

P-value: 0.9999989398972331

1-the P-value: 0.0000010601027669078212

my data below is:

54.53 52.78 57.09 51.72 53.58

54.97 53.77 56.81 52.75 58.68

54.05 53.67 52.27 51.74 54.65

Mean: 54.204

Standard Deviation:2.01878959492351

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

My random data is:

52.30 58.84 57.36 54.59 54.32

57.66 54.47 54.01 52.00 51.35

55.10 47.15 49.30 55.56 56.33

My mean for this sample is 54.022666666666666

My standard deviation for this data is 3.1709855760961023

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

$ME= 1.72913times(3.171/sqrt15)=1.416$

My confidence interval is: (54.023-1.416, 54.023+1.416) = (52.607, 55.484)

Hypothesis Test:

H0: μ = 50

HA: μ > 50

Standard error: $3.171/sqrt15= .8187$

Test Statistic: $(54.023-50)/(3.171/sqrt15) = 4.91389$

P-value: 0.000114This value is smaller than the .1 indicated from our confidence interval, so I reject the null hypothesis.

1-the P-value: 0.999886

My random data is:

54.24 50.00 52.48 54.62 55.04

55.61 53.36 50.54 55.26 48.94

53.60 61.62 57.68 56.19 50.36

Mean: 53.96933333333333

Standard Deviation: 3.2928157005965635

The t* multiplier for a 90% confidence interval is: 1.7613101151015698

The ME is: $1.7613*(3.2928/sqrt(15)) = 1.4975$

The confidence interval is: $[53.9693 - 1.4975, 53.9693 + 1.4975] = [52.4718, 55.4668]$

My standard error is: $3.2928/sqrt(15) = .8502$

My test statistic is: $(53.9693-50)/.8502 = 4.6687$

My p-value is: .9998

1-p: .0002

The above value is less than .1, so I reject the null hypothesis.

my data is:

56.31 59.00 55.52 54.24 56.08

51.51 57.73 54.09 45.90 53.60

52.53 55.26 55.45 49.12 56.69

mean: 54.2020

standard dev= 3.021

$ME= 1.72913times(3.02/sqrt15)=1.3093$

The confidence interval is (54.2020-.3.201, 54.2020+3.201 = (51.001,57.212)

H0:U=50

HA:U>50

My standard error is (3.021/sqrt(15))=1.058

My test statistic is (57.212−50)/(1.058)=2.589

My initial p-value for this data is 0.9892844500302621, however after subtracting this from 1, I get my true p-value of 0.0108

I reject the null hypothesis because my given value is smaller than 0.1

My random data is:

53.67 52.39 53.72 53.06 52.40

59.70 56.79 52.33 53.05 52.31

56.89 51.39 52.48 55.13 48.77

Mean: 53.605333

Standard Deviation: 2.629673

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

ME = 1.7613101151015698(2.629673/sqrt15) = 1.19589

my confidence interval is: (54.801, 52.409)

The setup for my Hypothesis test is below:

H0:u=50

Ha:u>50

Standard error: 2.629673/sqrt15= .678978

Test Statistic: (54.801-50)/(2.629673/sqrt15) = 7.0709

P-value: 0.9999972066612897

1-the P-value: 0.00000279333

My random data is:

55.81 55.44 55.09 54.07 53.83

57.11 53.11 58.35 50.62 46.07

57.26 53.28 57.33 47.75 55.26

Mean: 54.03

Standard Deviation: 3.39747

t* multiplier: 1.76131011

$ME = 1.76131 * (3.39747/sqrt15) = 1.54506161

Confidence Interval:

[54.03 - 1.54506, 54.03 + 1.54506] = [52.484, 55.5751]

Hypothesis Test:

H0: u = 50

Ha: u > 50

Standard Error:

$(3.397/sqrt15) = 0.8771$

Test Statistic:

$(54.03 - 50) / (3.397/sqrt15) = 4.59467$

P-value: 0.95643

1 - the P-Value: 0.04357

I reject the null hypothesis.

My random date is :

54.68 57.02 53.97 57.59 58.80

56.03 54.32 51.31 57.35 52.58

56.06 54.03 51.05 54.65 53.01

Mean:54.83

Standard Deviation:2.302579299084275

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

ME= 1.76131 x 2.3025/sqrt15 = 1.0471

My confidence interval is (54.83-1.0471), (54.83+1.0471) = (53.7829, 55.8771)

The setup for my hypothesis is below:

H0: u = 50

HA: u > 50

Standard error: 2.3025/sqrt15 = .594503

Test Statistic: (54.83-50)/(2.3025/sqrt15) = 8.10761

P-value: 9999994141064577

1-P-Value is: .000000585894

This value is smaller than the .1 indicated from our confidence interval so I reject the null hypothesis.

My data is:

53.89 57.88 47.60 57.31 54.31

56.18 54.95 54.65 52.38 54.18

57.57 53.36 51.01 54.38 56.17

Mean:54.388

Standard Deviation: 2.679

T* multiplier= 1.7613101151015698

ME=1.76131 x (2.679/sqrt15)=1.21832

My confidence interval is (54.388-1.21832, 54.388+1.21832)= (53.16968, 55.60632)

The set up for my confidence test is:

H0:u=50

HA:u>50

Standard error: .6917148

Test Statistic: (54.388-50) /.6917148=6.243654928

P value:.9999941329879616

1-minus P value= .0000059

The value is smaller then the .1 indicated from our confidence interval, so I reject the null hypothesis.

My random data is:

55.15 53.45 54.78 53.80 51.31

51.89 53.93 59.07 50.79 55.02

50.26 51.71 54.95 49.98 49.62

Mean: 53.04733333333333.

Standard Deviation: 2.5757286532626256.

For a 90% confidence interval, the t* multiplier is: 1.7613101151015698

$ME= 1.761 times (2.575/ sqrt15) = 1.14096992034$.

My confidence interval is: (53.04 -1.1409, 53.04 + 1.1409) = (51.899, 54.1809)

The setup for my hypothesis test is below:

H0:u=50

HA:u>50

My standard error = 2.575/ sqrt15 = 0.664862141099

Test Statistic: (53.04-50)/(2.575/ sqrt15) = 4.57237645533

p value= 0.945337181930095

1-the P-value: .0547

I want the P-value for u>50, so I calculate 1 - 0.945337181930095 for a P-Value of 0.0547. I reject my null hypothesis.

My data is below:

46.66 56.77 53.39 53.54 51.38

54.01 53.99 57.90 52.58 58.22

56.71 52.06 54.67 52.96 55.69

The standard deviation is: 2.9216821638683883

The mean is: 54.03533333333333

For a 90% confidence interval, the t* is: 1.7613101151015698

ME= 1.76131 x (2.92168/sqrt15) = 1.328687407

My confidence interval is: [54.03533-1.328687, 54.03533+1.328687] = [52.706643, 55.364017]

My hypothesis test is:

H0: u=50

Ha: u>50

Standard error: 2.92168/sqrt15= 0.7543745322

Test statistic: (54.03533-50)/(2.92168/sqrt15) = 5.34923944

P-value: 0.9999487024

In order for the value of u>50, I take 1-0.9999487024, and get 0.0000512976, and therefore I reject the null hypothesis.

my random data is:

53.83 54.43 54.21 56.29 51.41

55.04 51.65 53.80 52.80 55.77

50.97 56.54 58.73 46.63 53.24

mean:53.68933333333333

std dev=2.8645529859864363

for a 90% confidence interval, my t* multiplier is: 1.7613101151015698

my ME is: 1.7613101151015698 * (2.8645529859864363/sqrt(15)= 1.303

my confidence interval is:

(53.689 + 1.303), (53.689 -1.303)

the setup for my hypothesis test is:

H0:u=50

Ha:u>50

test statistic:

53.68933333-50/2.86455298/sqrt(15) =4.98812

p-value:0.999900575016809

1-the p-value: .00009943

The value is smaller then the .1 indicated from our confidence interval, so I reject the null hypothesis.

My random data is:

50.10 49.06 56.93 54.95 56.33

58.40 58.70 53.08 58.43 51.35

54.91 53.97 53.63 49.12 51.89

Mean = 54.056666666666665

Standard Deviation = 3.3021414408286778

For a 90% confidence interval, the t* is: 1.7613101151015698

ME= 1.76131 x (3.30214/sqrt15) = 1.50170855

My confidence interval is: [54.05666-1.50170, 54.05666+1.50170] = [52.55496, 55.55836]

The set up for my confidence test is:

H0:u=50

HA:u>50

Standard error: 3.30214/sqrt15= 0.85260

Test statistic: (54.05666-50)/(3.30214/sqrt15) = 4.7579

P-value: 0.9998470690091685

My p-value for the above test statistic is .9998470690091685, but I want the P-value for u>50, so I do (1-that p-value) to get 0.00015293099

That value is smaller than the .1 indicated from our confidence interval, so I reject the null hypothesis.