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Wize University Statistics Textbook > Hypothesis Testing with One Sample

Hypothesis Test for a Mean

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Hypothesis Test for a Mean

A hypothesis test may be performed for a population mean μ\mu. Be sure to review the hypothesis test steps before studying this section.

The test statistic is depends on 3 things:
  1. Whether the population standard deviation σ\sigma is known or not
  2. Whether the population distribution is normal or not
  3. Sample size

→ z\rightarrow\ z (if σ\sigma is known)
→ t\rightarrow\ t (if σ\sigma is unknown)







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Example: Hypothesis Test (Mean) - One-Sided Test


In 2013, the business department reported that the average monthly salary for co-op students is $2,960. Kenya randomly sampled 25 co-op students in the business department. The sample mean is $3,325 with a standard deviation of $940. Assume the population is normally distributed. At the significance level α=0.05\alpha=0.05, is there evidence that the monthly salary is higher now?

(a) Is this a z-test or a t-test?

This is a t-test because σ\sigma is unknown. Although the sample size is not very large, the population is assumed to be normal so we can proceed with the t-test.


(b) State the hypotheses:


Ho: μ=2960H_o:\ \mu=2960
Ha: μ>2960H_a:\ \mu>2960

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(c) What is the critical value?


df=n−1=25−1=24df=n-1=25-1=24
one-tail test ("greater than")
𝜶 = 𝟎.𝟎𝟓


CV=1.711CV=1.711 (t-table)



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(d) Find the test statistic:

t=x‾−μsn=3325−296094025=1.942t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}=\frac{3325-2960}{\frac{940}{\sqrt{25}}}=1.942



(e) Find the p-value.

t-table: 0.025<0.025< p-value <0.05<0.05
Software: p-value = 0.032


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(f) Select the correct conclusion:

i. We reject 𝐻o; there is evidence that the monthly salary is higher now.
ii. We reject 𝐻o; there is no evidence that the monthly salary is higher now.
iii. We fail to reject 𝐻o; there is evidence that the monthly salary is higher now.
iv. We fail to reject 𝐻o; there is no evidence that the monthly salary is higher now.

i. The p-value is smaller than the significance level (0.05). We reject HoH_o; conclude as per HaH_a: there is evidence that the monthly salary is higher now.

You can also say: The test statistic (1.942) is greater than the critical value (1.711). Reject Ho.
There is evidence that the average co-op salary is greater than $2,960.

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(g) What would be the conclusion if α=0.01\alpha=0.01?

i. We reject 𝐻o; there is no evidence that the monthly salary is higher now.
ii. We fail to reject 𝐻o; there is evidence that the monthly salary is higher now.
iii. We fail to reject 𝐻a; there is no evidence that the monthly salary is the same.
iv. We fail to reject 𝐻o; there is no evidence that the monthly salary is higher now.

iv. The p-value is greater than the significance level (0.01). Fail to reject HoH_o. (You do not reject HaH_a!)
There is no evidence that the average co-op salary is greater than $2,960.

If you are curious, the critical value at a = 0.01 is 2.492
The test statistic (1.9415) is smaller than the critical value (2.492). Fail to reject Ho.



(h) What does that say about the strength of the evidence?

Since we can reject Ho at a = 0.05 but not at a = 0.01, we can say that there is strong evidence but not very strong.

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Example: Hypothesis Test (Mean) - Two-Sided Test


The average time it takes to find a parking spot at the mall is 11.5 minutes. However, the mall's property was recently rezoned. Has the average time to find a parking spot changed since the rezoning? Based on a random sample of 41 cars, the average time for shoppers to find a parking spot is 10.25 minutes with a standard deviation of 4.5 minutes. The significance level is α=0.05\alpha=0.05.


(a) Is this a z-test or a t-test?

This is a t-test because σ\sigma is unknown. Although the sample size is not very large, the population is assumed to be normal so we can proceed with the test.


(b) State the hypotheses:

Ho: μ=11.5H_o:\ \mu=11.5
Ha: μ≠ 11.5H_a:\ \mu\neq\ 11.5

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(c) What is the critical value?


df=n−1=41−1=40df=n-1=41-1=40
two-tail test ("changed")
𝜶 = 𝟎.𝟎𝟓


CV=2.021CV=2.021 (t-table)


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(d) Find the test statistic:

t=x‾−μsn=10.25−11.54.541=−1.779t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}=\frac{10.25-11.5}{\frac{4.5}{\sqrt{41}}}=-1.779


(e) Find the p-value. (Remember: this is a two-tail test!)

t-table: 0.05<0.05< p-value <0.10<0.10 (be sure to find the correct p-value based on a two-tail test!)
Software: p-value = 0.0829


(f) Select the correct conclusion:

i. There is evidence that the average time to find a parking spot has decreased.
ii. There is evidence that the average time to find a parking spot has changed.
iii. There is evidence that the average time to find a parking spot has not changed.
iv. There is no evidence that the average time to find a parking spot has changed.


iv. The p-value is greater than the significance level (0.05). We fail to reject HoH_o, then we have no evidence for HaH_a.

(iii. is not correct - we didn't prove that HoH_o is true.)

The marketing department states that the average spending at the store is $335, but they believe that customers are spending less money, on average. A random sample of 31 customers reveal a mean of $302 with a standard deviation of $128.

(i) What are the correct hypotheses?
Extra Practice
Confidence Interval for a Mean
A random sample of 30 credit card statements showed an average balance of $1,200 with a standard deviation of $135.
Hypothesis Testing: One-Tailed Z-Test
Ronald the Cat is a popular English TV Series show with historical average of 1.2 million viewers per episode and a standard deviation of 0.41 million. Lately, the actor behind the character Ronald has made some controversial comments over the internet. Over 9 random episodes since the comments, the average number of viewers per episode dropped to 0.95 million. The producer is worried that the mean number of viewers has dropped.
a) State the null and alternative hypotheses.
b) Compute the test-statistic
Hypothesis Testing: One-tailed T-test for the Mean
Ryan's English teacher gave him 70% on his essay. He thinks he deserves more. The principal of his school randomly selects 7 English teachers to grade his paper The mean grade assigned is 80% with a standard deviation of 8%.
Hypothesis Testing: One-tailed T-test for the Mean
Ryan's English teacher gave him 70% on his essay. He thinks he deserves more. The principal of his school randomly selects 7 English teachers to grade his paper The mean grade assigned is 80% with a standard deviation of 8%.
Hypothesis Testing: One-Tailed Z-Test
Ronald the Cat is a popular English TV Series show with historical average of 1.2 million viewers per episode and a standard deviation of 0.41 million. Lately, the actor behind the character Ronald has made some controversial comments over the internet. Over 9 random episodes since the comments, the average number of viewers per episode dropped to 0.95 million. The producer is worried that the mean number of viewers has dropped.
a) State the null and alternative hypotheses.
b) Compute the test-statistic
Hypothesis Testing: One-Tailed Z-Test
Ronald the Cat is a popular English TV Series show with historical average of 1.2 million viewers per episode and a standard deviation of 0.41 million. Lately, the actor behind the character Ronald has made some controversial comments over the internet. Over 9 random episodes since the comments, the average number of viewers per episode dropped to 0.95 million. The producer is worried that the mean number of viewers has dropped.
a) State the null and alternative hypotheses.
b) Compute the test-statistic
Hypothesis test for a mean
Normal cows weigh an average of 50 tons. The cows in Hippy’s Farm have a gluten-free diet. Do their cows weight differently from normal cows? We randomly selected a sample of 41 cows from Hippy’s Farm; they weigh an average of 54.7 tons with a variance of 64. Use the Significance level of 0.01
Hypothesis Testing: One-tailed T-test for the Mean
Ryan's English teacher gave him 70% on his essay. He thinks he deserves more. The principal of his school randomly selects 7 English teachers to grade his paper The mean grade assigned is 80% with a standard deviation of 8%.
Hypothesis testing for a mean
In testing Ho:μH_o:\mu = 20 against Ha:μH_a:\mu > 20 the p-value is 0.32. The sample mean was 22.
Which of the following are true?
Hypothesis Test for a Mean
Normal cows weight an average of 50 tons with a standard deviation of 25tons. The cows in Hippy’s Farm have a gluten-free diet. Do their cows weight differently from normal cows? (Yes, No)
We randomly selected a sample of 41 cows from Hippy’s Farm; they weight an average of 55.7 tons.
Hypothesis testing for a mean
In testing H0:μH_0:\mu = 20 against H1:μH_1:\mu > 20 the p-value is 0.32. The sample mean was 22.
Which of the following are true?
Hypothesis testing for a mean
Goldilocks is a picky eater. The perfect temperature for porridge is 88°C. She won’t eat it if it’s too hot or too cold. Based on a random sample of 30 bowls of porridge, the average sample temperature is 91°C. The temperature is normally distributed with a standard deviation of 3°C. Is there evidence that the temperature of porridge is not to her liking?
Hypothesis Test for a Mean
After the Pokemon Go upgrade, we hypothesize that UBC students play the game for more minutes than 30 minutes a day.
Hypothesis test for a mean
You wish to test H0:μ=29   H1:μ≠29H_0:\mu=29\ \ \ H_1:\mu\ne29
A 90% confidence interval for the population mean is (22, 28)\left(22,\ 28\right)
Which of the following is not true?
Confidence Interval for a Mean
A random sample of 30 credit card statements showed an average balance of $1,200 with a standard deviation of $135.
Hypothesis testing: Mean
(iii) What is the p-value?
Hypothesis Testing: Mean
(ii) Find the test-statistic.
Hypothesis Test for a Mean
(iv) Using a 5% significance level, select the correct conclusion.
Hypothesis Test for a Mean
Parts (i) to (v) are based on the following information:
Mirco Management Ltd. allows full-time employees to take 1-hour lunch breaks per shift; however, they believe that they take more than an hour on average. If this is the case, the company will hire more floor supervisors and increase surveillance. A random sample of 30 employees revealed a mean of 64.5 minutes with a standard deviation of 11 minutes.
(i) Select the correct null and alternative hypotheses:
Hypothesis Test for a Mean
The professor wishes to see if the average grade exceeds 70%. His p-value is 0.0245. Select the correct statement(s). [There may be more than one correct answer.]
Hypothesis Testing for a Mean
Parts (i) to (iii) pertains the the following information:
The marketing department states that the average spending at the store is $335, but they believe that customers are spending less money, on average. A random sample of 31 customers reveal a mean of $368 with a standard deviation of $128.
(iii) At the 5% significance level, what is the correct conclusion?
Hypothesis Testing for a Mean
Parts (i) to (iii) pertains the the following information:
The marketing department states that the average spending at the store is $335, but they believe that customers are spending less money, on average. A random sample of 31 customers reveal a mean of $302 with a standard deviation of $128.
(ii) What is the p-value?
Hypothesis Test for a Mean
Parts (i) to (iii) pertains the the following information:
The marketing department states that the average spending at the store is $335, but they believe that customers are spending less money, on average. A random sample of 31 customers reveal a mean of $368 with a standard deviation of $128.
(i) What are the correct hypotheses?
Statistics: Definitions
One of the following statements is false. Which one is it?
Hypothesis Test for a Mean (one-tail)
In 2013, the business department reported that the average monthly salary for co-op students is $2,960. Kenya randomly sampled 25 co-op students in the business department. The sample mean is $3,325 with a standard deviation of $940. Assume the population is normally distributed. At the significance level 𝜶 = 𝟎.𝟎𝟓, is there evidence that the monthly salary is higher now?
(a) Is this a z-test or a t-test?
(b) State the hypotheses:
Hypothesis Test (Mean) - Two-Sided Test
The average time it takes to find a parking spot at the mall is 11.5 minutes. However, the mall's property was recently rezoned. Has the average time to find a parking spot changed since the rezoning? Based on a random sample of 41 cars, the average time for shoppers to find a parking spot is 10.25 minutes with a standard deviation of 4.5 minutes. The significance level is α=0.05\alpha=0.05.
(a) Is this a z-test or a t-test?
(b) State the hypotheses:
Statistical errors
For the following statements, select true or false. Enter T for True, F for False. (Use capital letters: T, F)
1. The mean weight of five cats is 20 lbs. We take away one that at weights 28 lbs. The mean weight of the remaining cats is 18 lbs.
T
2. The 95% confidence interval is [$800, $1,200]. The sample mean must be $1,000.
T