Testing hypotheses

Testing hypotheses

by Sianga Mutola -
Number of replies: 0

Q1: A school principal wants to test if it is true what teachers say – that high school juniors use the computer an average of 3.2 hours a day. What are our null and alternative hypotheses?

  • Null Hypothesis (H₀): The average time high school juniors use the computer is 3.2 hours per day.
    H0:μ=3.2H_0: \mu = 3.2H0​:μ=3.2

  • Alternative Hypothesis (H₁): The average time high school juniors use the computer is not 3.2 hours per day.
    H1:μ≠3.2H_1: \mu \neq 3.2H1​:μ=3.2

Q2: Duracell manufactures batteries that the CEO claims will last an average of 300 hours under normal use. A researcher randomly selected 20 batteries from the production line and tested these batteries. The tested batteries had a mean life span of 270 hours with a standard deviation of 50 hours. Do we have enough evidence to suggest that the claim of an average lifetime of 300 hours is false?

A. Hypotheses:

  • Null Hypothesis (H₀): The average lifetime of the batteries is 300 hours.
    H0:μ=300H_0: \mu = 300H0​:μ=300

  • Alternative Hypothesis (H₁): The average lifetime of the batteries is not 300 hours.
    H1:μ≠300H_1: \mu \neq 300H1​:μ=300

B. Statistical Test: Since the sample size is small (n = 20) and the population standard deviation is unknown, we would use a t-test to test the hypothesis.

C. Statistical Significance Level: Typically, a significance level of 0.05 is used. If the p-value is less than 0.05, we reject the null hypothesis.

  • Test Statistic (t):

    t=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}t=s/n​xˉ−μ0​​

    Where:

    • xˉ=270\bar{x} = 270xˉ=270 (sample mean)

    • μ0=300\mu_0 = 300μ0​=300 (population mean under null hypothesis)

    • s=50s = 50s=50 (sample standard deviation)

    • n=20n = 20n=20 (sample size)

    We would compute the test statistic and compare it to the critical value from the t-distribution with 19 degrees of freedom.

Q3: If the difference between the hypothesized population mean and the mean of the sample is large, we ___ the null hypothesis. If the difference between the hypothesized population mean and the mean of the sample is small, we ___ the null hypothesis.

  • Answer: If the difference is large, we reject the null hypothesis. If the difference is small, we fail to reject the null hypothesis.

Q4: At the Chrysler manufacturing plant, there is a part that is supposed to weigh precisely 19 pounds. The engineers take a sample of parts and want to know if they meet the weight specifications. What are our null and alternative hypotheses?

  • Null Hypothesis (H₀): The average weight of the parts is 19 pounds.
    H0:μ=19H_0: \mu = 19H0​:μ=19

  • Alternative Hypothesis (H₁): The average weight of the parts is not 19 pounds.
    H1:μ≠19H_1: \mu \neq 19H1​:μ=19

Q5: A group of students have an average SAT score of 1,020. From a random sample of 144 students, we find the average SAT score to be 1,100 with a standard deviation of 144. We want to know if these high school students are representative of the overall population. What are our null and alternative hypotheses?

  • Null Hypothesis (H₀): The mean SAT score of the students is 1,020.
    H0:μ=1020H_0: \mu = 1020H0​:μ=1020

  • Alternative Hypothesis (H₁): The mean SAT score of the students is not 1,020.
    H1:μ≠1020H_1: \mu \neq 1020H1​:μ=1020