Please can you answer these statistic hypothesis for me ?

Question

- A cords manufacturer claims that the standard deviation of the strength of his wrapping cord is 10 pounds. A sample of 14 wrapping cords produced a standard deviation of 12.4 pounds. At α=0.05, test the claim.

- A sample of 15 teachers from Los Angeles has an average salary of $35,270, with a standard deviation of $3,256. A sample of 20 teachers from New York has an average salary of $29,512, with a standard deviation of $1,432. Is there a significant difference in teachers’ salaries between the two states? Use α=0.05.

- Nationwide 13.7% of employed wage and salary workers are union members (down from 20.1% in 1983). A random sample of 300 local wage and salary workers showed that 50 belonged to a union. At α=0.05, is there sufficient evidence to conclude that the proportion of union membership differs from 13.7%?

- Nationwide, the average salary of registered nurses with more than 5 years experience is $65,470. A registered nurse wants to see how this compares with colleagues within her hospital. She checks the salaries of 15 registered nurses and finds the average salary to be $75,560 with a standard deviation of $10,270. Can she conclude that registered nurses in her hospital make more than the national average, using α=0.05?

- Stocks in market ABC are believed to be more stable than stocks in market XYZ. The indexes for markets ABC and XYZ have been tracked for 100 days. Although both indexes oscillated around the same level of 8000 points, the index of ABC recorded a standard deviation of 638 points while that of XYZ recorded a standard deviation of 819 points. At α=0.10, is there a significant difference between the variability of market ABC and XYZ.

- A recent survey of 200 households showed that 8 had a single male as the head of household. Forty years ago, a survey of 200 households showed that only 6 had a single male as the head of household. At α=0.05, can it be concluded that the proportion has changed?

Answer 1

 A cords manufacturer claims that the standard deviation of the strength of his wrapping cord is 10 pounds. A sample of 14 wrapping cords produced a standard deviation of 12.4 pounds. At α=0.05, test the claim.

Answer:

Here, we will use the chi-square test to test manufacturer's claim that the standard deviation is 10.

Ho :σ=10

H1: σ not equal to10

Sample size, n=14

sample standard deviation, s=12.4

degrees of freedom=(n-1)=14-1=13

alpha=.05.

Let's calculate the chi-squared test statistic

χ2=s^2*(n−1) /σ ^2​

Plugging in values from above,

χ2= 12.4^2*(14-1)/ 10 ^2 = 19.99

The critical chi squared value will for df=13, and alpha/2 since this is a two tailed test, so alpha splits equally in the two tails, so alpha/2=.025.

Lower critical value: 5.009

Upper critical value: 24.736

Since 5.009<19.99<24.7365.009, the calculated statistic falls in the non-critical region (acceptance region), so the decision is to not to reject the null hypothesis There is not enough evidence to conclude that the true standard deviation of the cord strength differs from the claimed 10 pounds. ​